A threshold in a continuous variable determines treatment assignment
Running variable (or “score”), X: a continuously distributed variable with a clearly defined cutoff (c) that determines which units are assigned to treatment and which ones are assigned to control.
Prior to the treatment, the outcome should not differ between the treatment and control group (continuity assumption). The distribution of the variable which indicates the threshold should have no jumps around this cutoff value.
Example using an arbitrary test score cut-off for being enrolled in tutoring programs on learning outcomes, from Andrew Heiss
sharp RDD: the threshold separates the treatment and control group exactly
fuzzy RDD: the threshold influences the probability of being treated. this is in fact an instrumental variable approach (estimating a Local Average Treatment Effect [LATE])
The value of the outcome (Y) for individuals just below the threshold is the missing counterfactual outcome. It increases continuously with the cutoff variable, as opposed to the treatment.
Here are some nice elaborations and fifures on these two variants of RDD from the World Bank
Sharp vs Fuzzy RDD illustration from the World Bank
In fuzzy designs, the probability of treatment is discontinuous at the cutoff, but not to the degree of a definitive 0 to 1 jump. For example, if food stamp eligibility is given to all households below a certain income, but not all households receive the food stamps, then the assignment rule defines treatment status probabilistically but not perfectly. Thus, the design is fuzzy. The fuzziness may result from imperfect compliance with the law/rule/program; imperfect implementation that treated some non-eligible units or neglected to treat some eligible units; spillover effects; or manipulation of the eligibility index.
A fuzzy design assumes that, in the absence of the assignment rule, some of those who take up the treatment would not have participated in the program. The eligibility index acts as a nudge. The subgroup that participates in a program due to the selection rule is called compliers (see e.g. Angrist and Imbens (1994), and Imbens, Angrist, and Rubin (1996)); under the RDD, the treatment effects are estimated only for the group of compliers."
With an RDD approach some assumptions can be tested. Individuals or units close to the threshold are nearly identical, except for characteristics which are affected by the treatment.
RDD’s strengths include that we can:
Load required packages. In this demo, we will use the package “rdrobust”
install.packages("rdrobust",
repos = "http://cran.us.r-project.org")
##
## The downloaded binary packages are in
## /var/folders/_c/txgkq4nd3pqfrsyx1ygqk9kh0000gq/T//RtmpYfx373/downloaded_packages
install.packages("rddensity",
repos = "http://cran.us.r-project.org")
##
## The downloaded binary packages are in
## /var/folders/_c/txgkq4nd3pqfrsyx1ygqk9kh0000gq/T//RtmpYfx373/downloaded_packages
library(tidyverse) # ggplot(), %>%, mutate(), and friends
# library(broom) # Convert models to data frames
library(rdrobust) # For robust nonparametric regression discontinuity
library(rddensity) # For nonparametric regression discontinuity density tests
library(modelsummary) # Create side-by-side regression tables
We are interested in the causal effect of fishing on crab mortality. This question is surprisingly hard to answer because it’s hard to tease apart natural mortality from fishing mortality, but it is an important effect to estimate for fisheries management.
### Make column for observation ID
crab_data <- data.frame(id = seq(1, 1000),
### Add columns for explanatory variables
### Add column for treatment variable
fished = c(rep(0, 500), rep(1, 500)),
### Add column for body size
size = c(runif(500, min = 10, max = 80),
runif(500, min = 60, max = 130)),
### And the rest of the covariates
sst = runif(1000, min = 10, max = 16),
salinity = runif(1000, min = 1, max = 10),
npp = runif(1000, min = 0, max = 1),
### And the error term
error = rnorm(1000, mean = 0, sd = 2))
### Visualize
ggplot(crab_data) +
geom_boxplot(aes(x = size, group = fished)) +
geom_vline(xintercept = 70)
### center size at legal catch size
crab_data <- crab_data %>%
mutate(size_centered = size - 70)
### Make column for outcome variable (total mortality)
crab_data <- crab_data %>%
mutate(natural_mortality = 10 + -0.4*size_centered + 2*sst + 0.3*npp + 1.5*salinity + error,
fishing_mortality = ifelse(size_centered < 0 & fished == 0,
10 + -0.1*size_centered,
ifelse(size_centered < 0 & fished == 1,
30 + 0.4*size_centered,
ifelse(size_centered > 0 & fished == 0,
10 + -0.1*size_centered,
30 + 0.4*size_centered))),
total_mortality = natural_mortality + fishing_mortality)
### Visualize mortality outcomes
ggplot(crab_data) +
geom_point(aes(x = size_centered,
y = natural_mortality,
color = fished))
ggplot(crab_data) +
geom_point(aes(x = size_centered,
y = fishing_mortality,
color = fished))
ggplot(crab_data) +
geom_point(aes(x = size_centered,
y = total_mortality,
color = fished))
### Visualize covariates
ggplot(crab_data) +
geom_point(aes(x = size_centered,
y = sst,
color = fished))
ggplot(crab_data) +
geom_point(aes(x = size_centered,
y = salinity,
color = fished))
ggplot(crab_data) +
geom_point(aes(x = size_centered,
y = npp,
color = fished))
Example: Rule-based example with cut off: fishing size limits of crabs
In order to be legally harvest, crabs must be larger than 70mm. Crabs smaller than 70mm are not legal to harvest. Harvesting is thus rule-based, according to size. Here, we would probably expect mortality to decline with body size, but the harvest rule changes this relationship at the discontinuity.
Since we know that the program was applied based on a rule, we next want to figure out how strictly the rule was applied (e.g. was there a lot of illegal harvest of crabs that were too small? Were all crabs over 70mm harvested? ).The threshold was 70 mm for legal harvest — e.g., did people who caught crabs that scored 68 mm slip through cracks? The easiest way to check this is with a graph, and we can get exact numbers to verify with a table.
ggplot(crab_data, aes(x = size_centered, y = fishing_mortality, color = fished)) +
# Make points small and semi-transparent since there are lots of them
geom_point(size = 0.5, alpha = 0.5,
position = position_jitter(width = 0, height = 0.25, seed = 1234)) +
# Add vertical line
geom_vline(xintercept = 0) +
# Add labels
labs(x = "Size (centered on 70 as cut-off)", y = "Fished or not") +
# Turn off the color legend, since it's redundant
guides(color = "none")
This discontinuity looks fuzzy rather than sharp We see some evidence of non-compliance around the size-based harvest rules. We will verify in Step 3.
Comparison of Fuzzy vs Sharp Discontinuities from Causal Inference: The MixTape by Cummingham
We next verify our conclusionss from the above visualization that the design is a fuzzy disconintuinity with a table examinging if there crabs under 70 mm harvested, and some over 70mm that weren’t.
crab_data %>%
group_by(fished, size_centered<= 0) %>%
summarize(count = n())
## `summarise()` has grouped output by 'fished'. You can override using the
## `.groups` argument.
## # A tibble: 4 × 3
## # Groups: fished [2]
## fished `size_centered <= 0` count
## <dbl> <lgl> <int>
## 1 0 FALSE 79
## 2 0 TRUE 421
## 3 1 FALSE 441
## 4 1 TRUE 59
Based on the table, we can conclude this is a fuzzy design becuase there are some crabs over the cut-off that were not fished, and some under that were fished.
Next, we need to see if any crabs self selected on either side of the size threshold; this seems unlikely. But, first, we’ll make a histogram of the running variable (size) and see if there are any big jumps around the threshold:
ggplot(crab_data, aes(x = size_centered, fill = fished)) +
geom_histogram(binwidth = 2, color = "white", boundary = 0) +
geom_vline(xintercept = 0) +
labs(x = "Size", y = "Count", fill = "Legal to fish")
## Warning: The following aesthetics were dropped during statistical transformation: fill
## ℹ This can happen when ggplot fails to infer the correct grouping structure in
## the data.
## ℹ Did you forget to specify a `group` aesthetic or to convert a numerical
## variable into a factor?
Here it doesn’t look like there’s a jump around the cutoff. There’s a tiny visible difference between the height of the bars right before and right after the 70-mm sie threshold (centered at 0), but it seems to follow the general shape of the overall distribution.
There’s a fuzzy discontinuity, but how big is it? And is it statistically significant?
This is likely a fuzzy RDD set up: some crabs < 70mm are probably harvested illegally, and fishers are not catching every crab > 70mm. So the treatment assigned is not necessarily the treatment received. To estimate the effect of fishing on total crab mortality in a fuzzy RDD, we take a two stage least squares approach (using a the fuzzy RDD as an IV.
Recall from Session 12 that instruments let us isolate causal effects for just compliers: they let us find the complier average causal effect, or CACE (the effect of a treatment for subjects/units who comply with the experimental treatment assigned to their sample group).
In this case, the instrument is fairly easy and straightforward: we create a variable that indicates if someone is above or below the threshold. That’s all. This variable essentially measures what should have happened rather than what actually happened.
Surprisingly, it meets all the qualifications of an instrument too:
First we will create the IV from the cut-off. We have already centered the running variable (size) on 0, so now we just need to create an indicator variable if the size if above or below 0.
##NEED TO FINISHED
## id fished size sst salinity npp error
## 1 1 0 41.84797 15.01488 5.545414 0.5122984664 3.3969839589
## 2 2 0 79.73803 10.72554 4.267921 0.8095724010 0.1253746350
## 3 3 0 55.19018 10.44435 3.126786 0.8203605369 4.3990486885
## 4 4 0 25.99801 10.57384 1.528624 0.7065953936 1.2974977910
## 5 5 0 16.93209 10.79003 3.185417 0.3172799677 0.5057971146
## 6 6 0 77.99594 10.96894 3.884679 0.9955174383 0.4577028855
## 7 7 0 75.78847 14.12099 8.459429 0.1811334651 0.3922578259
## 8 8 0 71.26553 14.02179 7.709277 0.3268148908 1.3296439782
## 9 9 0 14.34982 11.93525 4.715391 0.2017384777 -2.9739816639
## 10 10 0 22.80087 14.01309 7.572889 0.5894450329 -2.3019843168
## 11 11 0 42.60481 15.43312 1.602038 0.5618162802 3.2227520771
## 12 12 0 62.88149 15.71597 6.394025 0.8288963567 -2.1750029413
## 13 13 0 60.17699 10.35532 5.796374 0.4219403376 0.8822375574
## 14 14 0 32.59043 10.15416 3.818026 0.8700902204 -0.2491520273
## 15 15 0 51.26204 15.90615 4.762553 0.1247618941 2.7226801443
## 16 16 0 10.49994 13.33957 6.864297 0.6574287221 0.7003348420
## 17 17 0 42.11457 12.84864 9.428915 0.1309700655 0.6711096504
## 18 18 0 75.95654 14.13312 5.699696 0.7211235187 0.7889370497
## 19 19 0 40.02033 10.75790 2.766307 0.1889430520 1.5082758640
## 20 20 0 25.46802 14.32016 3.391951 0.7070721067 -0.3754883504
## 21 21 0 73.57829 11.65374 7.104121 0.2830847516 1.9474140003
## 22 22 0 73.77113 10.17380 1.659021 0.7839087658 1.4762403450
## 23 23 0 70.39111 10.72531 9.484125 0.1543507460 2.1068024106
## 24 24 0 48.51101 14.35260 8.366441 0.1691268147 -2.7909367857
## 25 25 0 24.51298 10.10666 2.818988 0.8754135838 -0.5422201475
## 26 26 0 77.88724 12.72474 6.710268 0.6226072919 -0.5719742850
## 27 27 0 65.55883 14.11271 7.756873 0.2401117554 -2.3161689689
## 28 28 0 67.41968 15.76721 6.684782 0.4999219819 3.0075259272
## 29 29 0 31.53491 11.77832 5.307226 0.7455336342 -1.9232422924
## 30 30 0 40.52941 12.73453 2.467462 0.7810508355 1.7487218861
## 31 31 0 19.45654 12.02799 9.415875 0.2963312902 0.9784903958
## 32 32 0 72.93853 10.68688 9.777817 0.0432374792 0.1140312712
## 33 33 0 62.42224 13.78415 4.352443 0.5107571867 0.6403206807
## 34 34 0 55.35832 12.60025 1.395540 0.3550621348 1.6030061242
## 35 35 0 66.14812 11.92191 5.977690 0.9392860658 0.0519689174
## 36 36 0 37.19295 13.58838 2.754159 0.1850668599 1.3820889683
## 37 37 0 62.96996 12.58675 8.341360 0.8649867503 2.6502074527
## 38 38 0 47.26075 10.51209 4.968321 0.7112351351 1.4034001158
## 39 39 0 15.21740 10.13269 3.673724 0.3645381115 0.2209358847
## 40 40 0 69.81422 13.23291 3.203782 0.9831388935 3.0807254861
## 41 41 0 32.60420 12.96117 1.522039 0.0617264931 0.1118555363
## 42 42 0 22.28027 14.43550 7.896019 0.1736047477 2.6013051240
## 43 43 0 16.39401 12.65178 5.314491 0.6231348314 -1.9095395440
## 44 44 0 26.70468 10.10301 4.972573 0.5215708176 0.6829802736
## 45 45 0 15.60613 14.31909 3.559101 0.4291872804 -2.7468414051
## 46 46 0 46.04189 15.84706 5.918764 0.1119348619 0.4560719601
## 47 47 0 29.03857 11.20811 2.234222 0.1252546546 -2.7555850172
## 48 48 0 12.56427 15.05049 4.436517 0.9253359330 1.3867785209
## 49 49 0 39.57454 11.12537 1.660280 0.5595281560 -1.2392024539
## 50 50 0 15.51346 11.31742 8.071734 0.9631643721 0.7381255821
## 51 51 0 48.38250 12.69948 8.017425 0.2620442947 0.8876103813
## 52 52 0 65.11413 15.76984 2.501628 0.0309782731 -0.3198516168
## 53 53 0 22.84824 15.15031 4.240319 0.0765342431 2.2165254546
## 54 54 0 64.20021 11.53364 2.128376 0.3658199674 -0.7852923782
## 55 55 0 14.50995 12.81946 3.799347 0.5117374037 -4.2985809929
## 56 56 0 18.21572 11.97044 5.108071 0.7054517686 2.5432584596
## 57 57 0 73.01384 11.20347 7.824564 0.1483750653 0.5144264427
## 58 58 0 79.01977 12.09899 1.066615 0.3810280049 -0.0683067826
## 59 59 0 28.77805 13.04654 9.829733 0.1768077554 -0.9312476481
## 60 60 0 35.24682 15.05166 4.142770 0.1817097727 -0.4606011425
## 61 61 0 70.20414 10.91182 7.219908 0.8815033613 2.4714938086
## 62 62 0 70.31827 13.46894 9.577780 0.7304476865 1.1759402509
## 63 63 0 57.06523 10.84315 3.655374 0.8600256352 -1.9380081427
## 64 64 0 72.91898 13.22309 8.883660 0.1794379631 -0.8443659858
## 65 65 0 38.67047 14.62388 3.753849 0.1743076388 -5.0651073159
## 66 66 0 69.97597 11.20178 1.063669 0.2996070071 1.5675415281
## 67 67 0 26.46307 11.63039 8.174687 0.7901809514 -1.1655098800
## 68 68 0 45.83037 10.30156 6.674546 0.5029937075 -0.0326504760
## 69 69 0 58.53935 10.55497 5.931420 0.0306283531 -0.1155761100
## 70 70 0 45.50775 13.11828 7.800584 0.0743867168 -2.0614441326
## 71 71 0 26.01266 12.91242 8.775044 0.3721621393 -0.8303960257
## 72 72 0 16.42100 11.29376 1.410725 0.2649603505 -2.4022119407
## 73 73 0 43.35972 15.23244 9.113410 0.8232880232 -3.2484080673
## 74 74 0 50.56978 15.68517 8.590039 0.7796957477 3.4820971738
## 75 75 0 65.27564 15.36784 3.018861 0.9696724361 1.7054502325
## 76 76 0 49.79546 15.89499 4.523502 0.5313015403 -0.0028462461
## 77 77 0 28.21531 10.98202 8.882294 0.9971313328 -2.4332206927
## 78 78 0 45.50296 12.67811 8.934493 0.8874159793 3.5915328205
## 79 79 0 22.60305 12.51546 6.139394 0.4119287438 -2.1188587402
## 80 80 0 56.39777 10.14164 1.298075 0.7278274598 0.5818998740
## 81 81 0 48.71824 12.92779 7.490425 0.6096635696 1.2272261547
## 82 82 0 26.68396 10.86137 5.355498 0.0612834888 2.0316798316
## 83 83 0 72.23963 12.44157 8.013368 0.8887657733 -0.6802550782
## 84 84 0 35.85515 12.88831 9.829259 0.8944283871 -2.2706815212
## 85 85 0 79.76453 12.60060 6.432885 0.9706627983 -3.1244579010
## 86 86 0 14.09337 12.05203 7.920201 0.5212248899 0.0667089679
## 87 87 0 33.37735 13.46307 1.760967 0.8986572046 0.9253533551
## 88 88 0 57.60634 13.89445 8.371074 0.9768107254 -0.4340780850
## 89 89 0 10.83747 10.21515 8.162783 0.4039448590 1.6791313729
## 90 90 0 24.49206 14.27505 1.577690 0.0234826030 0.9076616032
## 91 91 0 79.32137 13.17095 3.515679 0.9608223382 0.5264126248
## 92 92 0 30.44873 11.21197 6.909163 0.5110034277 -2.7297933247
## 93 93 0 33.52545 13.03120 1.188815 0.3936737247 -2.9302804428
## 94 94 0 78.79096 13.29567 9.266841 0.9073055424 0.7799769576
## 95 95 0 46.38155 12.87570 8.756405 0.3437323764 -1.4345715721
## 96 96 0 26.99989 14.09412 8.873921 0.2134562279 -1.6746520175
## 97 97 0 55.06387 12.80434 2.877051 0.7619143345 -2.9313977920
## 98 98 0 25.31338 15.31941 8.280588 0.2213303880 1.0348317305
## 99 99 0 67.39879 13.16029 3.749055 0.7366472881 -1.5175630351
## 100 100 0 49.89301 15.71041 8.294970 0.1869371529 -0.4131680964
## 101 101 0 38.66793 10.43607 4.144904 0.0392171801 -3.1838099014
## 102 102 0 14.68882 11.80734 6.164031 0.3123323028 0.3349038619
## 103 103 0 29.87938 14.65559 8.917850 0.9684240224 0.7865166947
## 104 104 0 25.41085 11.06034 4.393528 0.5109950609 -0.1353561202
## 105 105 0 15.57580 13.38560 2.307871 0.9150218209 -1.0580629055
## 106 106 0 47.39121 14.65672 6.960592 0.4558633366 2.4267994039
## 107 107 0 40.48732 14.71303 6.670824 0.5734539097 1.7028136421
## 108 108 0 36.26677 12.71401 2.170721 0.8508319431 -0.7982042586
## 109 109 0 13.34889 14.19463 6.777333 0.6029144635 1.7440143467
## 110 110 0 61.65646 13.21078 9.457444 0.9750003896 4.2425261966
## 111 111 0 28.79088 14.87107 2.544236 0.9093731826 -0.3275293352
## 112 112 0 10.06158 15.91344 4.996650 0.6399102909 1.1945664792
## 113 113 0 58.33446 12.34006 5.220775 0.6617895542 1.3830643760
## 114 114 0 34.10013 10.92581 2.290369 0.4201782672 -3.5480279999
## 115 115 0 57.30338 11.37179 6.177431 0.4678367155 1.1750156527
## 116 116 0 40.81362 13.88085 8.247046 0.9030558849 -3.8594159620
## 117 117 0 77.12404 15.20979 3.055968 0.7234343733 -0.8245310309
## 118 118 0 65.13657 15.48676 3.381153 0.9859646019 1.9353056903
## 119 119 0 73.22936 15.07964 1.471651 0.2880695490 0.5572993473
## 120 120 0 10.40575 11.91318 6.183794 0.4355266048 -2.6441643647
## 121 121 0 32.31772 10.79653 6.569972 0.7499470473 0.3139303127
## 122 122 0 57.41269 15.15925 5.322542 0.7455334389 1.7557478129
## 123 123 0 76.82869 11.16442 3.458440 0.5644706700 -0.0530984827
## 124 124 0 70.77481 13.85297 4.055676 0.7499118163 1.6598986400
## 125 125 0 38.63745 12.37969 4.872155 0.7665744072 -0.5987688536
## 126 126 0 11.02867 12.31414 2.113859 0.1527790776 -2.7719897802
## 127 127 0 50.34350 12.07887 1.184923 0.0673992594 1.0325604849
## 128 128 0 37.84255 12.84449 5.496648 0.6703295272 -0.1850263852
## 129 129 0 74.22559 11.71944 3.843540 0.0811288576 -0.4845985592
## 130 130 0 37.76976 12.77118 9.410009 0.9694322716 1.0104942167
## 131 131 0 33.71929 14.01138 9.654346 0.7572716225 0.0903822824
## 132 132 0 42.79985 10.70880 1.740517 0.9503490352 -2.3317726547
## 133 133 0 61.23944 10.24088 1.122617 0.2837087386 -0.5176609740
## 134 134 0 25.23067 13.44719 7.364621 0.7485409766 -0.3495062834
## 135 135 0 76.52360 15.83878 2.097535 0.5798271280 2.1719130269
## 136 136 0 28.28269 14.78496 8.763782 0.7496547576 -0.2857654708
## 137 137 0 12.21055 10.02816 2.486694 0.4247822030 0.8581857994
## 138 138 0 54.10035 12.98163 1.985439 0.5226041018 -1.9614800972
## 139 139 0 17.22799 15.29618 6.028626 0.9738611453 0.3716135322
## 140 140 0 42.40250 10.56270 1.400449 0.2197747743 -0.9425891016
## 141 141 0 30.80824 15.18755 4.983362 0.3841440172 0.1583607607
## 142 142 0 14.56892 12.83107 4.606661 0.1732167196 -0.5193953515
## 143 143 0 57.35698 13.75555 8.874459 0.4133129192 -1.3941493706
## 144 144 0 38.07288 10.16630 7.499890 0.8536553450 0.5192775206
## 145 145 0 36.98108 12.81230 3.734203 0.6272266400 0.1957090256
## 146 146 0 79.28549 13.39761 4.897393 0.4364014033 -0.3235363163
## 147 147 0 10.44172 13.74717 5.798280 0.0658066235 -1.5366326014
## 148 148 0 38.32252 15.38744 5.495818 0.1128019686 0.4300918557
## 149 149 0 56.75011 14.00072 7.443256 0.9207038709 2.0881601237
## 150 150 0 22.43354 11.36753 8.998676 0.2297360701 -1.0283498469
## 151 151 0 22.78758 14.53655 9.103666 0.2873810437 -1.2884900460
## 152 152 0 63.23957 14.68854 6.532568 0.7573658316 2.3955481533
## 153 153 0 26.93195 12.66553 8.064233 0.1063300585 -1.7965964186
## 154 154 0 75.14671 10.96817 5.406106 0.0701185360 -1.2751359919
## 155 155 0 74.31939 12.89253 8.772149 0.6503223164 -0.6606254659
## 156 156 0 15.78544 12.56871 3.785396 0.3949214439 -0.6447751623
## 157 157 0 55.39119 15.48601 5.857491 0.6279762271 -3.5255081747
## 158 158 0 28.28048 15.52833 7.092461 0.2378863627 1.2463038184
## 159 159 0 48.20912 11.87577 4.735966 0.2175597639 0.7327686184
## 160 160 0 37.09030 12.65978 5.525361 0.7789316403 -1.0029923669
## 161 161 0 52.79976 11.88368 3.458494 0.6415594837 1.7027603248
## 162 162 0 35.82119 15.20446 1.438349 0.0601551707 0.4280616658
## 163 163 0 64.53143 13.79527 4.743491 0.6878967229 3.7329732400
## 164 164 0 10.69172 13.48238 7.332244 0.4671270025 -2.5776046116
## 165 165 0 37.07448 14.15430 7.369232 0.5821742765 1.2168489754
## 166 166 0 76.37842 15.82521 9.739867 0.1195260952 -0.4268028459
## 167 167 0 32.28629 13.44353 4.865087 0.4803830616 -0.0063541646
## 168 168 0 24.51447 13.23236 2.487983 0.5654357602 0.1993445812
## 169 169 0 44.91922 10.18166 7.037157 0.8096255818 -2.7050220637
## 170 170 0 51.42400 15.18047 7.349462 0.1496962484 0.5504983468
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## 424 424 0 44.09390 15.35116 6.554568 0.0041635716 -0.2968673417
## 425 425 0 60.12215 15.70613 8.726366 0.6796121059 -2.4849569529
## 426 426 0 52.62165 14.49167 9.903337 0.8250616163 3.0293648107
## 427 427 0 65.45816 15.66282 3.083509 0.1729694637 1.1701675087
## 428 428 0 74.62324 14.27530 1.048396 0.5900472007 7.2121248190
## 429 429 0 64.05148 15.47070 8.355980 0.9967813599 2.4490258621
## 430 430 0 60.14832 13.03795 2.038228 0.8966289018 1.2269767135
## 431 431 0 41.06760 13.25776 5.586159 0.7707323357 -1.8698191655
## 432 432 0 21.17524 15.93643 4.053686 0.0898875613 -1.6162210797
## 433 433 0 19.39057 14.55736 7.376965 0.0997058915 0.8118020882
## 434 434 0 66.40170 11.60275 6.560765 0.9926928950 0.6406634879
## 435 435 0 35.55058 11.13460 6.128547 0.9752784087 0.8591499506
## 436 436 0 74.45650 14.62986 2.225857 0.6686680201 2.1569672305
## 437 437 0 75.07443 13.96170 1.477113 0.3937187307 -0.2825536193
## 438 438 0 55.77173 14.66111 3.301127 0.9772212701 -1.3624220059
## 439 439 0 45.33560 14.12431 2.000664 0.1370297200 0.6065689448
## 440 440 0 56.17164 12.07809 6.459881 0.5162824432 -0.7483572420
## 441 441 0 43.75993 12.66633 7.775599 0.3970756007 0.3580526783
## 442 442 0 49.68636 10.79552 3.368768 0.8308092637 -0.0388933922
## 443 443 0 60.94715 12.97481 9.730456 0.6021083628 -2.9711879588
## 444 444 0 76.87973 12.51879 7.363903 0.3398860849 -4.1295072207
## 445 445 0 21.80218 13.25431 4.330574 0.9273858250 -1.2551533636
## 446 446 0 46.87964 10.39933 3.144543 0.8698163386 -0.9710742953
## 447 447 0 30.20317 14.54391 4.939594 0.8982378046 -0.1637083344
## 448 448 0 59.79609 11.55675 4.586748 0.4634068774 -0.3843512451
## 449 449 0 68.90790 15.59354 6.781255 0.2159462769 -1.2368586133
## 450 450 0 73.31741 11.93059 6.378449 0.9466458680 -1.3123583848
## 451 451 0 32.02334 12.02544 7.072739 0.5304137829 -1.8051926603
## 452 452 0 57.25424 15.68416 2.350769 0.2741397829 -0.0079076274
## 453 453 0 50.56297 15.64910 4.484043 0.8429498901 -3.7051714452
## 454 454 0 52.31877 12.59731 9.644301 0.9617204498 -1.8339641199
## 455 455 0 32.71982 12.19022 9.625441 0.3475991657 -2.0952175663
## 456 456 0 49.17960 10.16150 5.164256 0.6551162121 2.0294600121
## 457 457 0 24.71140 14.72911 4.195559 0.1609277690 0.6806588270
## 458 458 0 10.90848 10.32208 2.011801 0.4738924245 0.1818382999
## 459 459 0 27.83821 13.72928 7.487488 0.8204968611 -3.2920896101
## 460 460 0 44.40471 11.90589 5.126837 0.5659135433 -1.6608637528
## 461 461 0 65.47207 12.86583 4.872788 0.8950806365 0.6419738748
## 462 462 0 75.86325 10.14427 2.051659 0.3737650225 -2.9365171080
## 463 463 0 16.83136 12.49160 9.706837 0.1146005511 1.8143734212
## 464 464 0 51.39573 12.00589 3.180673 0.9345218593 2.0625309267
## 465 465 0 28.26352 11.92784 7.773125 0.1256269335 2.3360033106
## 466 466 0 73.01473 15.04627 8.020581 0.7835060761 -2.9105275662
## 467 467 0 78.93526 15.59652 7.535608 0.9914062128 -0.5982757757
## 468 468 0 17.68080 11.14500 7.129912 0.9864133135 -0.5775421529
## 469 469 0 76.55174 11.52552 6.634586 0.2214863393 -4.0643515837
## 470 470 0 54.77879 10.00288 5.592868 0.5886946707 4.1220885865
## 471 471 0 62.46771 14.25536 4.331537 0.2695323895 -0.8923735243
## 472 472 0 20.01843 14.19031 8.092329 0.4363202138 -0.7204197007
## 473 473 0 54.96714 13.56013 8.411986 0.0003411265 0.1091861318
## 474 474 0 70.07682 12.81629 6.842217 0.3598351909 0.3659407057
## 475 475 0 43.51918 11.42335 1.292800 0.6596575389 0.7966076530
## 476 476 0 25.62279 15.43585 3.207972 0.1859156305 1.1288945075
## 477 477 0 16.28907 10.84736 7.090749 0.5313337154 -1.4055115025
## 478 478 0 44.25334 10.03680 1.160673 0.3142974034 -1.7580277527
## 479 479 0 20.86461 14.91444 4.555649 0.9801418099 -1.8546614081
## 480 480 0 11.87823 15.56351 3.401203 0.2837930175 -4.4868228982
## 481 481 0 33.15158 10.40613 4.147724 0.5786049184 1.0228941759
## 482 482 0 20.88009 15.50282 2.571700 0.0181206265 -1.1077485088
## 483 483 0 59.45803 10.81515 3.028410 0.7163714934 -0.2664437190
## 484 484 0 13.71196 10.89003 4.996990 0.3865434739 0.4793675262
## 485 485 0 41.00238 10.59809 2.623292 0.0510326747 -1.2309297984
## 486 486 0 70.79542 13.70722 5.301262 0.5883173903 0.4384610948
## 487 487 0 13.76568 15.50190 8.128257 0.7313689906 -1.9386659300
## 488 488 0 27.09578 11.64710 8.841020 0.7625001562 0.6035483622
## 489 489 0 20.67865 11.13734 5.559085 0.1945629921 -1.7174439049
## 490 490 0 58.46395 11.39980 9.888942 0.5747248929 0.8846338820
## 491 491 0 59.69128 14.69430 9.770612 0.7632682808 -0.7060349601
## 492 492 0 27.57357 13.63639 3.145035 0.4323440520 2.3714450780
## 493 493 0 51.88301 12.79097 3.541861 0.4758024579 0.8024162921
## 494 494 0 16.67719 14.26818 6.642059 0.0379647361 0.2760082018
## 495 495 0 45.13847 15.24529 6.825422 0.3084082138 3.0426899408
## 496 496 0 40.22571 10.05576 4.078273 0.7508620680 0.7123419172
## 497 497 0 77.96816 13.01885 9.885625 0.2146871290 -1.1354544050
## 498 498 0 73.55536 11.36350 8.492139 0.0485513110 0.0175468269
## 499 499 0 62.46098 12.63002 4.249147 0.8886069439 -1.4824810037
## 500 500 0 53.24974 13.62109 6.393217 0.1107497569 0.7517708133
## 501 501 1 71.09375 13.73441 3.224278 0.3811226401 -1.1277813367
## 502 502 1 126.32809 15.37795 9.350874 0.8336211659 -1.0800118494
## 503 503 1 60.70570 13.78599 9.442488 0.0485620485 -1.1495165868
## 504 504 1 111.51297 10.29176 4.928723 0.0173637520 1.6524307251
## 505 505 1 99.72688 11.99874 7.600097 0.0810297427 -0.4271348934
## 506 506 1 126.47106 10.69722 4.242467 0.9216875178 -0.9124521204
## 507 507 1 107.62346 11.44205 9.726791 0.5173406063 -1.6721363929
## 508 508 1 98.77177 12.48195 8.352858 0.4487846021 -2.5797907162
## 509 509 1 65.16515 15.05760 6.711653 0.1410812661 0.2326779029
## 510 510 1 77.10813 14.07526 6.088693 0.8098178387 -1.4209588861
## 511 511 1 90.69457 15.28851 5.838034 0.5742706414 -1.2770490721
## 512 512 1 81.12945 10.67494 3.415427 0.8947175797 0.6566267171
## 513 513 1 101.56572 14.99493 9.541247 0.1076762353 1.2859536159
## 514 514 1 98.35121 13.40496 1.881698 0.8619494471 -0.1146414082
## 515 515 1 105.74416 12.69122 2.864491 0.9931341421 0.4419941758
## 516 516 1 126.65594 15.12601 9.959940 0.4510069021 1.2504389498
## 517 517 1 88.09558 10.03126 7.264234 0.0960929722 -0.4829587033
## 518 518 1 90.82187 14.84416 2.530271 0.5234884126 4.0710212015
## 519 519 1 110.09624 11.96924 4.070851 0.1205681250 0.2777353969
## 520 520 1 122.90750 10.83117 9.156646 0.8867086505 -1.7720421094
## 521 521 1 83.56281 10.19737 6.471340 0.3720571841 -1.3236686607
## 522 522 1 105.39671 11.67692 5.469728 0.6124727714 -0.4078010704
## 523 523 1 124.72976 15.58337 7.809839 0.8692817960 2.5207817851
## 524 524 1 121.64639 11.94480 8.855619 0.3873691149 -2.9713829760
## 525 525 1 78.16460 14.95888 4.200646 0.2523658446 -0.8843918988
## 526 526 1 77.50127 12.40077 9.573604 0.5314695146 0.5287123658
## 527 527 1 81.66920 15.17542 1.421705 0.9759295133 0.2685532603
## 528 528 1 99.07612 11.38454 5.099804 0.3265321103 0.9626051063
## 529 529 1 89.99200 11.71025 8.268850 0.9289248106 -2.7824132262
## 530 530 1 87.86177 13.75883 6.309825 0.6556658014 1.0095248578
## 531 531 1 112.86845 13.62924 4.222356 0.5162548004 -2.1152462630
## 532 532 1 78.92798 13.34869 1.042287 0.2604255844 2.0960120682
## 533 533 1 83.26196 10.73944 2.299465 0.3967446918 -0.7314765868
## 534 534 1 119.68931 11.96477 2.061999 0.7454329457 0.7452390672
## 535 535 1 122.96620 14.95138 1.396510 0.3242380647 4.0801760678
## 536 536 1 113.31714 14.56473 9.948150 0.5355195485 -0.0003730287
## 537 537 1 124.65899 12.12800 2.558927 0.2557019340 0.2565894672
## 538 538 1 107.74554 14.46411 1.666547 0.7216503194 0.1279647112
## 539 539 1 63.08914 15.71141 7.483712 0.4272211366 0.2487158491
## 540 540 1 64.25090 10.36897 9.103643 0.3252407066 -3.5686421788
## 541 541 1 126.52824 14.88484 8.588881 0.9679650776 1.0598105852
## 542 542 1 110.30468 10.90103 9.042107 0.4297251031 -1.8461421821
## 543 543 1 102.70844 14.04437 8.290922 0.2720003361 1.1061963104
## 544 544 1 76.82632 10.21477 3.579597 0.6440374067 0.4717897051
## 545 545 1 117.89912 13.98916 9.537609 0.5797165108 -2.5611940841
## 546 546 1 62.72415 15.72086 4.476041 0.0923021757 -0.5645884724
## 547 547 1 120.72582 13.57795 6.426918 0.9414535225 0.5790043059
## 548 548 1 61.94241 12.69431 8.807178 0.0808953515 0.5464023607
## 549 549 1 101.37838 14.22506 7.837434 0.1470729166 0.9914475961
## 550 550 1 126.11002 11.10884 8.412291 0.5867140444 0.4016734988
## 551 551 1 82.27595 13.94485 5.521372 0.4820108393 -3.9801003541
## 552 552 1 127.51614 11.26688 6.821205 0.0976113854 0.6344551657
## 553 553 1 88.80640 10.08139 1.568284 0.8277146965 3.5057969212
## 554 554 1 81.85504 14.40172 9.394055 0.9935439401 -1.6133422251
## 555 555 1 110.59072 15.44455 5.905034 0.7245926908 -1.1422829725
## 556 556 1 77.84199 12.28347 5.401996 0.4055178768 0.2939166806
## 557 557 1 104.98405 11.43773 5.747301 0.4479928722 3.5246118323
## 558 558 1 98.24844 14.57829 7.128312 0.6596412850 -0.9155837244
## 559 559 1 76.70357 12.98858 3.968288 0.6551032648 -2.3418918831
## 560 560 1 92.85246 14.47199 6.241274 0.6268161251 -3.1686730524
## 561 561 1 65.24394 13.95940 1.233960 0.9402681233 -2.6937091899
## 562 562 1 80.85783 12.51497 5.959051 0.6813228996 2.4595293331
## 563 563 1 111.17503 13.74410 1.618870 0.9198924657 -2.9422017169
## 564 564 1 107.20313 11.24045 7.284155 0.8554095225 -0.4774261729
## 565 565 1 62.07709 10.65758 8.713307 0.7329774878 2.2212352093
## 566 566 1 125.97037 13.53590 5.255857 0.7950108610 0.0233511323
## 567 567 1 119.63912 13.59427 3.898774 0.0546908062 3.0608445056
## 568 568 1 128.81516 15.72543 9.625129 0.3064621857 0.6514285762
## 569 569 1 86.85478 13.30193 9.673018 0.1163828149 -0.1826875431
## 570 570 1 60.41396 13.21590 3.933433 0.6578448047 -0.3677385493
## 571 571 1 72.58332 11.08750 5.450847 0.8892998572 0.9995442162
## 572 572 1 68.99734 13.90499 1.259839 0.7746068556 4.0109144291
## 573 573 1 114.54196 12.16096 1.318244 0.2454774850 1.0883967457
## 574 574 1 88.81161 14.69879 1.912856 0.9838066960 0.2162441426
## 575 575 1 65.08443 11.28499 6.063479 0.2250178438 -2.1948531830
## 576 576 1 125.85078 15.67139 8.121180 0.9676186277 2.5025869106
## 577 577 1 127.97222 13.01875 9.981692 0.9479246957 -0.1670800841
## 578 578 1 92.53388 12.90727 6.775118 0.6409801200 -1.5661874921
## 579 579 1 126.76222 14.30024 4.114406 0.3227941853 -1.9315885164
## 580 580 1 114.45424 12.69667 8.943676 0.4509644352 -2.6162277915
## 581 581 1 105.89943 12.95519 4.775822 0.8744527786 1.7481987635
## 582 582 1 113.98057 11.19931 1.756662 0.4741021350 -1.0757211357
## 583 583 1 102.06807 13.48389 6.514471 0.5019704988 -1.6963186372
## 584 584 1 103.65743 11.00417 5.840404 0.7209678893 0.8265054281
## 585 585 1 77.87212 13.95456 7.483491 0.2110914073 1.1781853858
## 586 586 1 123.55098 14.73105 1.423518 0.2947722888 -1.5549017369
## 587 587 1 65.67367 10.22679 3.415892 0.6387341635 0.6146724349
## 588 588 1 78.44929 12.48415 9.808110 0.9299567665 -1.0666057764
## 589 589 1 113.52809 14.42400 2.927018 0.8122926936 2.7639029451
## 590 590 1 129.11221 10.83874 9.282479 0.4221205253 -2.2015218435
## 591 591 1 71.90160 14.60120 2.608383 0.5024504336 2.7763084618
## 592 592 1 120.04748 12.16184 5.467234 0.9282543513 -0.0950758972
## 593 593 1 79.87857 10.46516 1.523942 0.9885493203 0.8585133182
## 594 594 1 113.91810 12.83868 6.091970 0.1090847554 0.3999940532
## 595 595 1 68.22396 10.47893 5.898581 0.9876280774 -0.4097475938
## 596 596 1 79.86288 15.53500 4.166497 0.1678521822 3.2820979483
## 597 597 1 61.19947 13.47359 7.010138 0.6176363213 -0.6985686881
## 598 598 1 73.41503 10.15375 5.939332 0.4154718057 -4.1694487910
## 599 599 1 127.31210 15.17727 9.386552 0.2197215504 3.4722830531
## 600 600 1 82.04856 12.17720 4.129320 0.7828368268 -1.9322223834
## 601 601 1 122.89147 10.22953 3.884780 0.7611336571 -3.2469200740
## 602 602 1 90.70649 14.76079 3.424879 0.1978355844 1.3707186597
## 603 603 1 62.44121 14.92982 7.541224 0.9821627806 -1.8182524220
## 604 604 1 73.98503 14.12738 8.255691 0.6073207150 -0.5640998663
## 605 605 1 102.80856 13.61256 3.840115 0.6271756294 0.4299693751
## 606 606 1 78.23796 11.99844 5.764991 0.8748706898 1.6962616682
## 607 607 1 85.86202 13.32467 6.689241 0.8628359661 -3.4729835705
## 608 608 1 124.16979 11.95920 2.240594 0.8901314882 -0.4951265332
## 609 609 1 99.03590 11.46940 4.952784 0.1583011772 -4.6749685926
## 610 610 1 91.87050 14.04735 1.616860 0.9132679559 1.0297170118
## 611 611 1 128.95299 14.19806 2.588168 0.7595219722 -0.0317307380
## 612 612 1 60.63628 13.11654 3.702142 0.0089792863 0.0848083537
## 613 613 1 66.20274 12.74622 3.966828 0.7719678674 0.2389427859
## 614 614 1 81.92418 15.00601 7.560038 0.1737857335 1.1217697746
## 615 615 1 103.22342 10.19367 2.872438 0.6535111961 0.4761029831
## 616 616 1 113.32461 15.52728 6.073652 0.1819786888 -0.0774222570
## 617 617 1 113.66937 12.33438 4.289158 0.5588189340 -0.3725501409
## 618 618 1 86.98113 11.36934 2.144613 0.7425258609 -1.7684092809
## 619 619 1 118.43302 14.89077 4.839411 0.2663916922 2.0304083114
## 620 620 1 75.67531 15.99457 4.614154 0.1274022085 -0.6376629291
## 621 621 1 84.73703 11.40081 5.071217 0.9775445643 0.5091754800
## 622 622 1 97.37698 10.72696 4.573730 0.7640859266 -0.1560813318
## 623 623 1 129.26840 11.13548 7.992133 0.6951857314 -2.0054916665
## 624 624 1 104.62107 11.76567 9.085490 0.8324761062 1.2290657377
## 625 625 1 91.76325 10.95237 1.657323 0.8555798288 -2.9648819466
## 626 626 1 75.21960 14.93207 8.367494 0.7970403242 -2.2114269508
## 627 627 1 103.19470 11.12863 7.597325 0.4762284488 2.8579739902
## 628 628 1 71.14733 12.96985 3.799667 0.7595792820 -1.0105729989
## 629 629 1 127.56881 13.00589 2.371670 0.4974276884 0.6353450579
## 630 630 1 87.26262 13.02898 6.455686 0.1869858657 -0.6331183873
## 631 631 1 64.41432 13.30734 6.355570 0.4928022348 -1.0048365036
## 632 632 1 128.16355 10.15181 2.262575 0.0040129796 0.1338179441
## 633 633 1 63.61155 15.27585 1.370792 0.2512427124 3.1404716553
## 634 634 1 90.95011 12.08355 2.346858 0.6747299551 1.7406924674
## 635 635 1 117.89657 12.79735 9.692960 0.2627696262 0.4307791789
## 636 636 1 77.68609 15.41417 5.932440 0.8888018322 0.5169252500
## 637 637 1 95.03188 13.17728 3.822366 0.8555523343 0.5613992638
## 638 638 1 72.11967 13.57216 1.644029 0.0046312015 -3.7798527414
## 639 639 1 86.57895 10.92480 3.482500 0.6735297786 -2.3836786467
## 640 640 1 127.97412 12.50735 5.501585 0.5503835045 -3.5960453774
## 641 641 1 112.30355 14.11468 7.211773 0.2090930198 -2.5031554201
## 642 642 1 73.90254 11.41225 4.301113 0.3099311965 -1.4031644349
## 643 643 1 103.63438 10.67842 4.988927 0.5553910141 -0.1960718462
## 644 644 1 117.53617 15.20117 9.712668 0.3215704684 -5.6958455510
## 645 645 1 73.16716 13.06031 4.539068 0.6787757871 -1.1394611760
## 646 646 1 93.22770 14.38620 1.577585 0.8903287775 -1.1277282325
## 647 647 1 68.06833 14.66192 9.944449 0.1355676162 1.5299553866
## 648 648 1 94.14657 15.66798 8.574401 0.3348297004 -2.1923807064
## 649 649 1 75.46969 12.48481 9.564024 0.4361352012 -1.7867926767
## 650 650 1 117.73631 15.64146 2.399623 0.1520993349 -0.3466060378
## 651 651 1 78.25068 12.63499 4.139099 0.1738081744 -1.7959963858
## 652 652 1 114.59001 12.85813 2.261998 0.5716521444 -1.5400911379
## 653 653 1 101.99263 10.22411 3.947036 0.3560642302 0.8034361595
## 654 654 1 114.07889 10.97910 4.175316 0.8196217325 -2.3490228909
## 655 655 1 115.39233 11.72523 4.585496 0.8671832110 -2.7482801129
## 656 656 1 125.45707 13.12836 7.656521 0.3827806769 2.1630526092
## 657 657 1 88.14015 11.99829 4.389754 0.1456623056 -0.7280714006
## 658 658 1 73.37135 14.39776 1.684903 0.7700847795 0.8638167765
## 659 659 1 92.05532 15.93471 1.474193 0.2374253576 0.8894896648
## 660 660 1 112.74128 12.03237 2.272816 0.6034437628 -0.6019003992
## 661 661 1 118.87836 10.90377 2.162548 0.7938392861 -2.4776662958
## 662 662 1 101.77763 15.74620 2.774411 0.6678117164 -1.1374923837
## 663 663 1 86.73386 12.65505 9.412207 0.9229583535 4.7445116686
## 664 664 1 94.59739 13.47288 8.898343 0.5525978496 -0.0657536253
## 665 665 1 109.83213 11.41326 9.449637 0.2220031919 -1.7328901640
## 666 666 1 128.67814 13.52263 3.474435 0.7584428091 2.1185061855
## 667 667 1 121.58862 14.96588 2.239605 0.1283352373 -3.2514737019
## 668 668 1 125.49574 14.45059 1.366447 0.6841582113 2.3641220385
## 669 669 1 106.04468 12.67405 1.914319 0.7617197733 1.8735693874
## 670 670 1 90.82412 14.45647 9.105721 0.5588631153 -0.8343830565
## 671 671 1 117.36964 13.02610 4.513783 0.1763611413 -1.1878926340
## 672 672 1 75.27841 13.26637 7.420659 0.7203251712 2.1677600223
## 673 673 1 128.87775 11.82290 5.076439 0.8224302437 1.1882196270
## 674 674 1 104.42799 14.39172 1.723633 0.2770354059 1.9049068251
## 675 675 1 60.73863 14.19311 9.279399 0.0484020461 0.9046795473
## 676 676 1 84.03755 12.66196 9.512154 0.8787558530 1.0585488745
## 677 677 1 84.66854 10.89828 8.150108 0.4813408155 -3.4081121416
## 678 678 1 117.80104 13.44284 6.950090 0.7553483481 1.4504276857
## 679 679 1 60.74533 15.78355 3.881187 0.9935191753 1.0605579453
## 680 680 1 81.67254 13.44599 2.679285 0.7830495222 -3.5098830728
## 681 681 1 62.25209 15.21333 5.400361 0.4150345083 0.6358955431
## 682 682 1 118.30088 12.87195 7.062045 0.6235786092 1.2465327176
## 683 683 1 91.88426 10.35207 4.859515 0.7273605629 -2.2387670011
## 684 684 1 95.00512 15.04798 4.396901 0.1512960664 -0.8497282446
## 685 685 1 77.37011 15.46457 2.980307 0.5207782558 0.7784986831
## 686 686 1 64.31016 14.50060 4.610657 0.9018138973 -1.1501567462
## 687 687 1 85.35310 13.78461 4.549883 0.3497820836 -0.4250373844
## 688 688 1 101.74763 13.44986 2.673678 0.3343934503 -1.6282804978
## 689 689 1 90.52439 11.39923 5.035895 0.2047378493 -0.7053422182
## 690 690 1 70.64950 13.74313 4.367925 0.1999019468 0.1797921260
## 691 691 1 105.67126 10.84119 2.562103 0.2277797519 0.8307588598
## 692 692 1 90.75400 12.85485 5.042407 0.0904976553 0.8012975068
## 693 693 1 75.62299 10.54072 8.502436 0.3807463341 -0.0727553636
## 694 694 1 66.16380 13.58043 3.939569 0.3377748833 1.5956781790
## 695 695 1 99.09358 11.57802 2.751146 0.3273471666 -1.3484343314
## 696 696 1 95.99445 15.49400 3.214185 0.2579937803 0.5544009581
## 697 697 1 108.42121 10.15390 5.728883 0.7976588311 0.2183880883
## 698 698 1 74.58613 15.43237 2.094772 0.0681968592 0.2940315100
## 699 699 1 96.06878 12.12093 9.098548 0.8341346914 -2.2595392425
## 700 700 1 77.97835 15.12533 9.394104 0.7797295759 0.8426044301
## 701 701 1 120.96790 15.06921 4.775708 0.4939177795 1.3452776595
## 702 702 1 83.33325 15.30763 4.530343 0.4847417087 0.2956032912
## 703 703 1 91.90062 10.51839 8.027868 0.0605730237 -2.2269745061
## 704 704 1 126.57396 11.05689 4.110621 0.8503145932 3.8479837099
## 705 705 1 81.21869 15.59641 2.621131 0.7973399092 0.3021538351
## 706 706 1 123.14577 10.47015 6.986632 0.7746588963 1.9942552137
## 707 707 1 72.72763 11.49872 8.294926 0.9113698122 -1.2412466843
## 708 708 1 108.14784 13.70138 5.687487 0.0435425350 1.1805913518
## 709 709 1 93.04247 10.94106 3.971770 0.2491262159 -3.3234002817
## 710 710 1 83.73825 13.79641 1.493518 0.8713187645 0.3052080728
## 711 711 1 83.17460 13.15491 3.171608 0.9821822916 0.7061510303
## 712 712 1 91.80312 15.20661 8.763561 0.2077412487 2.8074845155
## 713 713 1 124.48839 10.91239 7.562919 0.9343241509 0.5440337334
## 714 714 1 115.81172 11.59413 8.991550 0.4305132201 -2.1306073269
## 715 715 1 91.24145 15.28978 9.369576 0.0745322369 2.2533183523
## 716 716 1 127.68318 14.30989 6.871591 0.2402927922 2.6850338857
## 717 717 1 112.68098 10.89020 2.901394 0.0784283669 -1.9111954758
## 718 718 1 101.08457 15.30609 5.962340 0.3888930355 -0.8916601043
## 719 719 1 115.31076 15.33232 6.467308 0.1211602176 -1.0759012942
## 720 720 1 126.28719 11.70984 9.588379 0.5414245313 0.6607392455
## 721 721 1 123.81912 12.80468 7.511002 0.3488636706 0.0745062826
## 722 722 1 77.39054 15.05622 4.947266 0.7906360670 0.7120185584
## 723 723 1 116.84576 12.78222 7.311625 0.8467163986 1.1011484964
## 724 724 1 81.20085 15.02889 3.618348 0.9407222481 -0.9136612054
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## 726 726 1 81.92156 13.94442 1.638076 0.0634890683 2.7222137692
## 727 727 1 90.56563 12.57577 5.898903 0.0284334728 -2.2545883812
## 728 728 1 70.76862 10.35851 5.706363 0.4452790804 -2.2664028216
## 729 729 1 115.31251 15.14741 3.406150 0.6360279582 -0.8732262294
## 730 730 1 120.59893 10.09933 5.139453 0.4367471812 -1.8067311378
## 731 731 1 70.45134 14.38207 1.832447 0.8313998200 1.2516768788
## 732 732 1 110.65553 12.67180 9.078351 0.3344134788 -2.2028186797
## 733 733 1 121.51828 11.43801 9.361825 0.8673747515 -0.0767220656
## 734 734 1 117.18381 11.16589 3.787514 0.0885740428 0.5748809206
## 735 735 1 80.60761 13.15036 4.502737 0.5477184905 -0.6441491497
## 736 736 1 71.50789 11.85746 1.420443 0.8396393417 2.3202609062
## 737 737 1 125.91954 12.83866 2.205540 0.5141616028 -0.7529075569
## 738 738 1 74.18923 11.56498 7.107955 0.2379039100 1.2152466466
## 739 739 1 113.79396 12.91568 5.012497 0.9023067565 -0.3791664615
## 740 740 1 117.88156 13.92753 5.358877 0.2162190655 0.6195332093
## 741 741 1 80.42343 13.72521 2.931174 0.2822762718 -0.1222810949
## 742 742 1 65.88584 14.96211 2.559213 0.9062963116 0.3095557226
## 743 743 1 109.26618 12.68112 3.624593 0.1142329862 -2.1834231169
## 744 744 1 126.47545 13.26018 2.740696 0.9131322429 -2.2737748822
## 745 745 1 91.19318 12.92716 6.772593 0.0780302463 -1.3953665737
## 746 746 1 126.39357 14.92083 3.329297 0.6302305562 2.9016909843
## 747 747 1 127.62313 12.51275 2.532563 0.9769565023 1.6557653948
## 748 748 1 106.76317 10.56556 7.335490 0.6698698299 1.6104098930
## 749 749 1 125.09534 14.77543 4.240642 0.7636017499 2.8132133085
## 750 750 1 87.83891 13.68554 9.495127 0.3119425455 -1.0601716316
## 751 751 1 60.27105 12.39097 7.686155 0.2333122594 0.8097925471
## 752 752 1 99.85678 10.22363 4.098246 0.3478666586 -0.5563097386
## 753 753 1 64.74139 15.01595 6.550680 0.6191551220 2.0596254079
## 754 754 1 67.37266 10.78387 7.265543 0.5076509153 1.1454328389
## 755 755 1 92.95084 10.08584 1.410210 0.2035401447 3.1856837047
## 756 756 1 75.39162 15.45032 7.163066 0.4041950197 -4.4208359703
## 757 757 1 96.01163 11.00417 5.906387 0.6045738717 1.2724709117
## 758 758 1 68.46913 12.72527 6.304188 0.4330587834 -0.4913755079
## 759 759 1 88.12140 14.87703 5.720698 0.8176870900 0.6287511182
## 760 760 1 112.96867 11.23562 2.377429 0.8236676033 1.0246333094
## 761 761 1 111.91951 13.27487 5.364757 0.1566935449 2.0719683604
## 762 762 1 92.19133 13.96229 9.670211 0.9774178448 1.7750924228
## 763 763 1 100.74303 14.20989 9.593393 0.0645644146 0.0813698437
## 764 764 1 114.32064 10.68773 8.273775 0.9849790223 -2.6888605693
## 765 765 1 125.06143 12.24429 1.632132 0.0604971559 -1.0859874682
## 766 766 1 103.70969 12.89142 5.125113 0.1017172730 0.4195786119
## 767 767 1 64.59106 13.91322 5.276980 0.3318891332 -3.6679920561
## 768 768 1 75.23825 10.81800 4.450981 0.3178508370 -0.2952296468
## 769 769 1 101.79324 14.80426 7.422849 0.2061930094 1.5389415258
## 770 770 1 127.20311 11.84489 8.164508 0.1625500340 2.2114417622
## 771 771 1 73.66398 15.58306 1.862108 0.0193893774 -2.5715873958
## 772 772 1 96.99224 12.99637 5.854310 0.9413217974 -0.5081173674
## 773 773 1 101.29451 11.50823 7.389711 0.3911684838 -3.7793186740
## 774 774 1 94.22564 15.77714 3.647901 0.9678436790 1.8769998972
## 775 775 1 107.81657 15.66376 4.591316 0.1666842627 2.2910489156
## 776 776 1 90.21312 10.03248 5.610960 0.6123270872 0.8745474941
## 777 777 1 83.69636 12.55900 6.747173 0.1726096771 -0.7542645514
## 778 778 1 83.23125 14.86456 6.413297 0.7625138371 2.0116701719
## 779 779 1 67.03180 11.38445 2.736147 0.6905585239 -2.1986112913
## 780 780 1 108.83379 10.78142 8.107913 0.2595457942 -1.4857003831
## 781 781 1 90.16483 15.60871 3.591699 0.2113246762 -0.8462211749
## 782 782 1 122.50342 10.43665 8.256013 0.3982881645 -0.4156856434
## 783 783 1 96.28344 11.61515 9.787375 0.3187180622 -2.4187319510
## 784 784 1 126.62151 10.20898 4.573368 0.0075759876 2.0844811559
## 785 785 1 124.34863 11.53864 1.331378 0.1035330025 3.1636251523
## 786 786 1 100.61046 14.97924 4.020214 0.9156274013 0.6787734778
## 787 787 1 68.77793 10.23553 1.905364 0.0246044686 -2.9389304033
## 788 788 1 108.37788 13.12932 4.834754 0.4731256606 -0.1822501750
## 789 789 1 79.14754 13.27917 5.598132 0.3460128228 1.6842626624
## 790 790 1 116.47985 12.48641 5.497747 0.5027486840 -2.5038198132
## 791 791 1 106.40346 12.81383 5.527549 0.9255413855 0.6074556252
## 792 792 1 86.20887 11.66404 7.046780 0.4103619480 -1.7785932165
## 793 793 1 75.98902 13.87569 7.936954 0.3428693130 -1.9277166155
## 794 794 1 89.96464 10.40215 5.492007 0.6547787974 -0.0012034403
## 795 795 1 105.88686 13.65570 4.884320 0.1415023287 -0.2135737161
## 796 796 1 61.70159 11.11899 6.733630 0.5713806637 -1.1992227526
## 797 797 1 123.01168 10.37212 2.354061 0.3096808170 2.0000457087
## 798 798 1 127.23841 12.02772 5.566346 0.9314062106 -0.1026715620
## 799 799 1 78.82574 10.68667 3.936392 0.5444165596 0.3077376357
## 800 800 1 74.56045 14.33181 3.161112 0.4347603484 -0.8755362581
## 801 801 1 103.04756 13.61288 7.238580 0.8669892396 -1.2230288790
## 802 802 1 90.23729 10.38456 5.504120 0.2024095280 -0.2652672491
## 803 803 1 103.09829 14.77487 6.279299 0.5898167060 -3.1083640991
## 804 804 1 60.65577 11.43607 9.749152 0.7252991649 0.4748140525
## 805 805 1 113.69684 11.26409 1.608408 0.1194482700 1.8804115423
## 806 806 1 61.37126 10.27167 2.950859 0.3174538191 3.7010034117
## 807 807 1 120.18826 14.75629 8.863163 0.4298888594 1.3786734296
## 808 808 1 67.25077 10.25433 8.759610 0.6241753534 1.8459798248
## 809 809 1 63.21590 10.17846 1.831199 0.7842971964 -3.9719621357
## 810 810 1 65.09407 10.39706 8.210039 0.6825516310 0.1210419373
## 811 811 1 114.59318 10.14178 3.883834 0.5315038546 4.0242210509
## 812 812 1 77.13524 15.87034 4.506286 0.6899188310 0.5397521830
## 813 813 1 99.36500 14.02010 5.500532 0.4517551535 0.4124024934
## 814 814 1 108.87158 10.87220 9.893421 0.4595525835 0.9907693500
## 815 815 1 80.42127 15.60530 4.682741 0.9407198040 0.5435824007
## 816 816 1 114.28437 10.13735 6.086894 0.0426531816 -0.1980367109
## 817 817 1 76.73024 13.85878 2.009422 0.9896889930 1.5841679721
## 818 818 1 121.21070 12.98531 8.783619 0.1055218072 2.3929867331
## 819 819 1 77.27374 11.00620 2.188616 0.3504635149 -0.5152033023
## 820 820 1 129.53908 13.62537 8.909686 0.0935812439 0.7519342930
## 821 821 1 96.08745 15.31468 6.538444 0.5327423175 -2.9976096982
## 822 822 1 70.18645 13.23568 8.604733 0.2126297709 4.1332282071
## 823 823 1 61.47107 15.30044 4.080464 0.6754926480 0.3377128803
## 824 824 1 105.63517 12.44862 1.618994 0.4047276599 0.3047882194
## 825 825 1 97.71137 13.64662 8.217352 0.4126807321 0.9059592039
## 826 826 1 115.98743 14.39911 3.749829 0.9684331408 -2.4236570331
## 827 827 1 83.87986 13.34409 6.035016 0.8067069575 -0.2994612873
## 828 828 1 93.22830 14.18047 6.515390 0.9142884610 4.1050377917
## 829 829 1 64.06907 10.34899 4.332669 0.4721457071 -1.6652304396
## 830 830 1 68.56895 15.72803 9.625133 0.0564645401 -5.1113885458
## 831 831 1 77.55588 14.42155 5.560088 0.5032336442 2.4975727192
## 832 832 1 108.93512 11.38385 1.356874 0.6352648062 1.4488203438
## 833 833 1 109.29548 12.37215 6.592784 0.6839837197 -0.0940903635
## 834 834 1 96.39777 15.57890 2.556138 0.9081668651 -4.4366393246
## 835 835 1 96.97179 11.32585 5.774142 0.4443721652 -0.3992019413
## 836 836 1 106.77901 14.42406 5.919855 0.3380089903 -0.3382573968
## 837 837 1 72.84836 14.62182 9.804443 0.6171011208 -2.1288114226
## 838 838 1 116.97003 11.58655 8.971104 0.0549857740 -4.6227126873
## 839 839 1 74.20985 10.21168 8.914843 0.1507632060 -1.3763962995
## 840 840 1 96.92385 13.32468 2.053596 0.7684549892 -0.2794063498
## 841 841 1 69.84830 12.46388 1.979853 0.7525645487 -0.3649881107
## 842 842 1 127.85779 11.90338 4.921515 0.8348243576 -2.0622866870
## 843 843 1 104.64889 10.26605 7.558790 0.2743188040 1.3894132496
## 844 844 1 81.66816 12.92866 9.663810 0.2920360020 0.6273714177
## 845 845 1 104.68647 13.87309 5.969815 0.2304270824 -0.5943148122
## 846 846 1 91.47189 13.22779 7.726067 0.0227801919 0.6133857999
## 847 847 1 65.53870 15.56224 2.720228 0.0150705371 0.5899821156
## 848 848 1 129.00069 11.00713 5.031286 0.3535467375 1.0000464119
## 849 849 1 103.70789 12.00660 6.194293 0.6394315553 -2.1441499182
## 850 850 1 90.60185 14.22740 6.850071 0.4172186912 0.8563020981
## 851 851 1 114.64248 15.55674 9.063164 0.0209517733 1.6288513318
## 852 852 1 105.25225 10.04498 6.808241 0.1414858256 -3.0315069773
## 853 853 1 73.90359 11.67162 8.869080 0.8067720123 -4.7153049260
## 854 854 1 79.38880 13.08088 2.719555 0.5712682677 -1.7543625622
## 855 855 1 96.60399 12.03886 8.029391 0.8302758995 -1.3400605331
## 856 856 1 111.33236 12.70335 8.527125 0.7756017228 2.9109991493
## 857 857 1 118.94764 11.35659 5.192489 0.6426293838 -1.3255957182
## 858 858 1 124.88408 12.90453 6.783822 0.8395085728 2.3082244573
## 859 859 1 87.05293 10.76100 5.687115 0.2121135490 1.5913805543
## 860 860 1 111.78319 14.29005 8.277761 0.3151510174 2.2167607526
## 861 861 1 92.68646 15.62644 4.242102 0.6502662452 2.4378404524
## 862 862 1 86.35151 11.90323 2.372962 0.0155342680 -0.2159303953
## 863 863 1 105.93922 12.47210 6.445525 0.5657894390 -3.3238926091
## 864 864 1 100.19617 13.37891 3.479403 0.8989611943 1.3530014003
## 865 865 1 122.76925 14.21679 4.410105 0.9856144793 -0.7317918926
## 866 866 1 106.31799 14.66117 8.696831 0.8993575317 -1.6911112426
## 867 867 1 127.19361 10.86034 5.495156 0.7015993781 -1.6244742357
## 868 868 1 86.19069 15.67866 4.774034 0.5579089997 1.2243236922
## 869 869 1 95.59941 14.24013 2.360709 0.1581420209 0.6700088439
## 870 870 1 74.51388 10.91308 7.708526 0.8259845907 -0.8431860356
## 871 871 1 64.43735 11.07598 4.166244 0.6841743798 -0.4208812539
## 872 872 1 107.98331 13.97003 5.984136 0.4291387927 -0.0549393447
## 873 873 1 118.27705 12.70050 5.385417 0.9632956251 2.2894868187
## 874 874 1 111.72915 14.77709 4.254063 0.1636654702 0.9892619831
## 875 875 1 74.12670 13.03378 9.711435 0.4315551273 -0.6509221979
## 876 876 1 108.27296 12.92587 9.432193 0.8689757821 -1.6932189072
## 877 877 1 70.92381 12.71766 5.495539 0.6847803425 -1.8978846319
## 878 878 1 123.51517 12.57391 1.659225 0.7898183372 0.1541436508
## 879 879 1 103.03165 10.28767 5.455962 0.3719889789 -2.6062134052
## 880 880 1 70.03016 13.89464 8.203285 0.9975404097 -3.1849963911
## 881 881 1 106.80789 12.89588 1.114615 0.9528138617 1.0135934114
## 882 882 1 64.73123 11.45578 6.616902 0.9237347397 3.2824919708
## 883 883 1 119.29346 14.49262 8.517682 0.4056376314 1.3051484412
## 884 884 1 108.69345 11.99251 7.256328 0.6392724365 -1.0655708101
## 885 885 1 123.35904 11.81098 4.735285 0.7956720612 0.6086694165
## 886 886 1 93.95671 11.97898 1.730128 0.9969724836 -1.2945421216
## 887 887 1 128.99368 15.18829 6.038215 0.4148474743 -1.8370869537
## 888 888 1 96.64733 10.15426 7.926085 0.4397394927 -1.1509806475
## 889 889 1 108.98181 13.99794 7.751809 0.1993018554 0.3329667333
## 890 890 1 94.71827 12.24099 5.599176 0.7997770982 -2.3511533536
## 891 891 1 120.70019 15.59468 8.019638 0.3968722234 -1.5730254820
## 892 892 1 106.36639 13.92319 6.523463 0.7781228938 1.6239023928
## 893 893 1 102.63239 12.79297 5.860014 0.5898978186 2.7203915041
## 894 894 1 72.63083 11.82394 2.869431 0.5063792102 -0.3038405296
## 895 895 1 88.43415 11.55829 2.615742 0.3561927096 -2.7497631909
## 896 896 1 101.82796 11.56131 9.833745 0.9238783447 0.6040778580
## 897 897 1 67.25759 12.27861 4.589680 0.0677326338 -0.1444883537
## 898 898 1 129.73182 13.55151 1.370922 0.4628555330 -0.1455461295
## 899 899 1 110.94521 10.55461 5.070351 0.5125620537 0.6568316433
## 900 900 1 107.72422 15.73940 2.886229 0.8220927296 0.2079047062
## 901 901 1 112.21568 13.95345 5.486004 0.2332717043 2.6990489919
## 902 902 1 96.83041 10.66543 3.214323 0.1656776094 1.3064243819
## 903 903 1 105.43787 14.55392 3.411749 0.7707665192 1.8471542910
## 904 904 1 71.90629 12.51874 9.949316 0.5504564345 -2.1430557406
## 905 905 1 108.31261 13.61917 3.815743 0.2780532511 2.2228056379
## 906 906 1 101.01957 15.45406 3.386632 0.4239236799 0.5605311797
## 907 907 1 60.19315 12.34444 2.353853 0.0493758747 1.0524375413
## 908 908 1 109.32569 11.74253 6.995425 0.6465675947 0.2411211085
## 909 909 1 122.24687 10.44747 7.452284 0.4239445932 -0.7850421919
## 910 910 1 110.90773 12.43620 3.044645 0.8123932581 -1.4875101610
## 911 911 1 98.98280 11.54426 7.683160 0.6342692652 0.7843334517
## 912 912 1 124.17878 12.92696 3.106524 0.6520189301 -1.8688794208
## 913 913 1 102.10790 11.25958 6.750556 0.0106751563 -0.4334017928
## 914 914 1 124.63678 10.90601 5.556459 0.7617144666 -1.1733490455
## 915 915 1 60.95895 14.81319 5.344424 0.3204094048 -0.5127666478
## 916 916 1 72.95584 13.01927 5.004287 0.6506150973 -1.9880331557
## 917 917 1 75.65147 10.82894 4.924244 0.7463110066 2.4806263710
## 918 918 1 112.92436 13.01274 6.680388 0.9921975811 -2.0062611353
## 919 919 1 123.72942 11.04655 9.247372 0.6650407785 -1.9498350423
## 920 920 1 121.86737 14.69692 1.327561 0.5598950004 -1.1583522884
## 921 921 1 95.82485 13.91634 4.836299 0.9590055747 1.1041679242
## 922 922 1 95.54569 15.52681 8.421412 0.0211880857 0.3398467668
## 923 923 1 126.94678 12.80013 6.282144 0.6029123126 -2.1581765502
## 924 924 1 122.75353 15.43095 6.090838 0.2631914383 -0.1555298772
## 925 925 1 117.53787 13.00611 9.721841 0.2507068235 0.4696031392
## 926 926 1 73.13532 11.73671 9.482929 0.4781717146 2.8548461496
## 927 927 1 119.07623 13.03274 3.215020 0.0714573015 -3.1742873327
## 928 928 1 89.38015 14.35385 8.128842 0.4046197988 -2.1712038703
## 929 929 1 89.80057 11.30643 1.747814 0.2674128397 -0.5248261313
## 930 930 1 102.84492 11.62128 4.270279 0.9258768272 3.0127100099
## 931 931 1 66.51858 14.49753 6.233716 0.6565893402 -2.3368490461
## 932 932 1 67.34877 14.90068 7.985992 0.8190235386 -3.2831816381
## 933 933 1 110.30445 10.74669 7.257351 0.9546628832 -0.2222008216
## 934 934 1 92.35198 13.81905 2.584770 0.7458657927 -0.9771096680
## 935 935 1 69.38605 14.89276 3.273833 0.6618956525 -0.0519652853
## 936 936 1 71.51008 10.20110 4.001558 0.8282731059 3.1810893440
## 937 937 1 84.64076 12.38066 4.413688 0.3682986819 -2.5218669874
## 938 938 1 91.98762 12.33534 4.694032 0.3902498137 0.4197280514
## 939 939 1 74.82872 11.44689 9.607515 0.0763875970 -0.1364098430
## 940 940 1 73.88750 11.93869 2.381694 0.4127557455 -1.3072442070
## 941 941 1 67.06045 12.12979 4.230165 0.8024664996 -2.1428456072
## 942 942 1 121.19039 12.28839 4.136287 0.2154391315 -1.7739147931
## 943 943 1 121.51981 14.64885 4.162589 0.9980353138 -0.8388920597
## 944 944 1 109.12508 11.26791 1.366690 0.4498045584 4.0249342631
## 945 945 1 102.47449 10.48794 5.720330 0.4422812464 0.8174487967
## 946 946 1 123.68943 14.85840 3.176927 0.0434377382 4.2689973814
## 947 947 1 96.51747 12.88361 2.743403 0.7265394859 -0.4778035238
## 948 948 1 116.98921 14.01699 1.005104 0.6513465634 -3.2342598931
## 949 949 1 74.30882 10.59169 8.049193 0.9129053173 1.2054256124
## 950 950 1 96.46387 14.50453 4.958689 0.3402656501 -2.8514918618
## 951 951 1 85.33481 14.55324 5.625418 0.9730518488 -2.1673835389
## 952 952 1 69.45870 13.15271 7.473152 0.0634970439 1.9066691420
## 953 953 1 67.06190 11.33790 1.009375 0.4816362192 -2.5677827635
## 954 954 1 80.96301 14.06491 5.812191 0.1910568762 1.7819644684
## 955 955 1 71.24477 14.81386 9.220477 0.9967272545 -1.0080177245
## 956 956 1 64.17507 10.65176 2.718660 0.1790875897 -0.8697094361
## 957 957 1 126.50618 11.26048 4.440940 0.7913353522 -0.4806180259
## 958 958 1 112.87603 10.71037 7.065820 0.2989096609 -1.7246511037
## 959 959 1 71.08046 11.24802 5.941208 0.2845480100 -0.0319534007
## 960 960 1 102.52673 12.01703 6.606123 0.0421712250 -5.5356256417
## 961 961 1 102.51417 10.10904 9.863845 0.7293606452 0.4464276409
## 962 962 1 119.39540 11.36439 4.785019 0.9721764128 0.0177559392
## 963 963 1 92.52473 13.89096 1.952146 0.7568617254 -1.3909757964
## 964 964 1 103.65286 13.34701 8.842640 0.3359481078 3.2903410948
## 965 965 1 97.99204 13.69119 8.850792 0.9964917498 0.2630992463
## 966 966 1 77.02587 15.74889 5.321142 0.8294452664 -2.7128709776
## 967 967 1 78.38257 10.75087 4.742583 0.9529253186 -3.2980871293
## 968 968 1 122.66292 15.73787 7.974426 0.1396756303 -0.1757628238
## 969 969 1 127.42031 12.89283 7.313428 0.5842473472 2.9990931974
## 970 970 1 75.07706 12.66910 7.169634 0.2242697459 -2.8885069013
## 971 971 1 87.78635 15.76022 5.929715 0.6931879267 -2.7909455763
## 972 972 1 122.19044 12.41388 2.491964 0.6558009095 3.4980865438
## 973 973 1 109.90379 13.53608 3.059641 0.0738397685 -4.4044064732
## 974 974 1 81.45692 12.37662 5.439390 0.3746809852 -0.9408532934
## 975 975 1 110.39187 14.84879 4.851423 0.9011173144 -3.1615097369
## 976 976 1 110.38594 12.73479 8.136902 0.2016765780 -1.6325972682
## 977 977 1 76.84486 12.36896 7.032631 0.5741789762 1.1241698505
## 978 978 1 77.31594 15.14398 7.579990 0.3313007588 -0.8543313754
## 979 979 1 74.22767 13.58545 2.147123 0.3010862097 3.0128046921
## 980 980 1 77.01145 10.11523 9.417747 0.2384989471 -1.9421217497
## 981 981 1 93.64225 10.44021 3.879255 0.6555605242 2.3544427885
## 982 982 1 120.91979 13.14897 3.176853 0.6207212261 0.2648822435
## 983 983 1 77.89268 13.02247 2.231900 0.1668624172 -1.6969894734
## 984 984 1 94.91745 11.04675 8.563649 0.3212920043 -0.9738471310
## 985 985 1 103.10992 11.85934 2.452476 0.7801170920 -1.1302943187
## 986 986 1 117.36385 15.97449 3.825748 0.7465923594 1.8569017709
## 987 987 1 119.73978 13.78413 1.002572 0.2402267612 -3.5074006066
## 988 988 1 85.93774 13.14335 3.759584 0.3072390859 -1.9176750086
## 989 989 1 103.24354 13.82810 7.764957 0.0414880360 -1.0774093729
## 990 990 1 60.72923 10.74540 7.561107 0.5253195420 3.3730351171
## 991 991 1 94.90496 12.93416 6.781721 0.9968742153 3.5061253797
## 992 992 1 82.13074 11.33939 9.275298 0.7938808908 1.9667898968
## 993 993 1 114.29683 10.90511 3.657062 0.8973897772 -0.3510245384
## 994 994 1 84.45683 13.18158 1.687644 0.1906968970 1.0267294824
## 995 995 1 63.70844 14.60221 3.799787 0.8020162454 5.3355377772
## 996 996 1 82.37184 10.31899 5.716730 0.4440468918 -2.2989068764
## 997 997 1 125.33845 15.54976 4.131872 0.5687901990 0.8186723602
## 998 998 1 66.64182 11.25565 8.198279 0.6582111607 1.4435924914
## 999 999 1 111.40925 12.41786 5.430706 0.0177538309 1.5004109088
## 1000 1000 1 109.81406 13.50030 6.209681 0.6986729356 0.0697279733
## size_centered natural_mortality fishing_mortality total_mortality
## 1 -28.15202814 63.15937 12.815203 75.97458
## 2 9.73802694 34.32599 9.026197 43.35219
## 3 -14.80981793 46.14795 11.480982 57.62894
## 4 -44.00198895 52.55088 14.400199 66.95108
## 5 -53.06790654 58.18634 15.306791 73.49313
## 6 7.99593676 35.32289 9.200406 44.52329
## 7 5.78847281 49.06234 9.421153 58.48349
## 8 1.26553241 50.52898 9.873447 60.40242
## 9 -55.65018068 60.29019 15.565018 75.85521
## 10 -47.19913332 66.14002 14.719913 80.85993
## 11 -27.39518695 57.61867 12.739519 70.35819
## 12 -7.11851308 51.94405 10.711851 62.65590
## 13 -9.82301458 44.34322 10.982301 55.32552
## 14 -37.40956812 51.01106 13.740957 64.75202
## 15 -18.73796023 59.21142 11.873796 71.08522
## 16 -59.50006015 71.67317 15.950006 87.62318
## 17 -27.88542670 61.70522 12.788543 74.49376
## 18 5.95653962 45.43845 9.404346 54.84280
## 19 -29.97966936 49.22209 12.997967 62.22005
## 20 -44.53197982 61.37767 14.453198 75.83087
## 21 3.57829048 44.56468 9.642171 54.20685
## 22 3.77113494 33.03910 9.622887 42.66198
## 23 0.39111078 47.67347 9.960889 57.63436
## 24 -21.48898532 57.11025 12.148899 69.25915
## 25 -45.48701716 52.35702 14.548702 66.90572
## 26 7.88723896 41.97479 9.211276 51.18607
## 27 -4.44116585 49.39305 10.444117 59.83717
## 28 -2.58032347 55.75123 10.258032 66.00926
## 29 -38.46509352 55.20394 13.846509 69.05045
## 30 -29.47059447 52.94153 12.947059 65.88859
## 31 -50.54346428 69.46457 15.054346 84.51891
## 32 2.93853257 44.99207 9.706147 54.69821
## 33 -7.57775693 47.92161 10.757776 58.67938
## 34 -14.64168056 44.86001 11.464168 56.32418
## 35 -3.85188379 44.68486 10.385188 55.07005
## 36 -32.80705266 55.86843 13.280705 69.14913
## 37 -7.03004105 53.40726 10.703004 64.11026
## 38 -22.73925059 49.18913 12.273925 61.46306
## 39 -54.78259602 58.01931 15.478260 73.49757
## 40 -0.18577813 44.72147 10.018578 54.74005
## 41 -37.39579936 53.29409 13.739580 67.03367
## 42 -47.71972981 72.45630 14.771973 87.22827
## 43 -53.60599381 62.99508 15.360599 78.35568
## 44 -43.29531692 55.82246 14.329532 70.15199
## 45 -54.39387111 63.11630 15.439387 78.55569
## 46 -23.95810687 60.64516 12.395811 73.04097
## 47 -40.96142628 49.43411 14.096143 63.53026
## 48 -57.43572895 71.39443 15.743573 87.13801
## 49 -30.42546002 45.84000 13.042546 58.88255
## 50 -54.48653892 67.56412 15.448654 83.01278
## 51 -21.61749583 57.03832 12.161750 69.20007
## 52 -4.88587265 46.93592 10.488587 57.42451
## 53 -47.15175600 67.76128 14.715176 82.47645
## 54 -5.79978781 37.90422 10.579979 48.48420
## 55 -55.49004926 59.38890 15.549005 74.93790
## 56 -51.78427556 65.07160 15.178428 80.25002
## 57 3.01383919 43.49719 9.698616 53.19581
## 58 9.01977136 32.23600 9.098023 41.33402
## 59 -41.22194966 66.44825 14.122195 80.57045
## 60 -34.75317537 59.81267 13.475318 73.28798
## 61 0.20414375 45.30780 9.979586 55.28738
## 62 0.31826792 52.57231 9.968173 62.54048
## 63 -12.93477402 40.66326 11.293477 51.95674
## 64 2.91898229 47.81354 9.708102 57.52164
## 65 -31.32953272 52.39754 13.132953 65.53049
## 66 -0.02402870 35.66609 10.002403 45.66849
## 67 -43.53692958 62.00912 14.353693 76.36282
## 68 -24.16963140 50.40104 12.416963 62.81801
## 69 -11.46065338 44.48495 11.146065 55.63101
## 70 -24.49224963 55.69521 12.449225 68.14444
## 71 -43.98733754 65.86360 14.398734 80.26233
## 72 -53.57899859 53.81248 15.357900 69.17038
## 73 -26.64028460 61.78969 12.664028 74.45372
## 74 -19.43021676 65.74349 11.943022 77.68651
## 75 -4.72436075 49.15007 10.472436 59.62250
## 76 -20.20454316 56.81359 12.020454 68.83404
## 77 -41.78469197 59.86728 14.178469 74.04575
## 78 -24.49704107 62.41454 12.449704 74.86425
## 79 -47.39695419 61.20351 14.739695 75.94320
## 80 -13.60222616 38.47154 11.360223 49.83176
## 81 -21.28175770 57.01405 12.128176 69.14223
## 82 -43.31603968 59.13247 14.331604 73.46407
## 83 2.23963026 45.59372 9.776037 55.36975
## 84 -34.14484815 62.17610 13.414485 75.59058
## 85 9.76453328 38.11146 9.023547 47.13501
## 86 -55.90663416 68.57009 15.590663 84.16075
## 87 -36.62264726 55.41159 13.662265 69.07386
## 88 -12.39365911 55.16194 11.239366 66.40131
## 89 -59.16252666 68.13981 15.916253 84.05606
## 90 -45.50793759 60.03452 14.550794 74.58531
## 91 9.32136579 38.70153 9.067863 47.76939
## 92 -39.55127220 56.03171 13.955127 69.98683
## 93 -36.47455369 49.62326 13.647455 63.27072
## 94 8.79096380 48.02738 9.120904 57.14829
## 95 -23.61844745 57.00194 12.361845 69.36379
## 96 -43.00011044 67.08855 14.300011 81.38856
## 97 -14.93612675 43.19588 11.493613 54.68949
## 98 -44.68661625 72.03557 14.468662 86.50423
## 99 -2.60121093 41.68809 10.260121 51.94821
## 100 -20.10698515 61.54899 12.010699 73.55969
## 101 -31.33207443 46.45028 13.133207 59.58349
## 102 -55.31118121 65.41380 15.531118 80.94492
## 103 -40.12062259 69.81325 14.012062 83.82532
## 104 -44.58914897 56.56458 14.458915 71.02349
## 105 -54.42420427 61.21914 15.442420 76.66156
## 106 -22.60879108 61.36140 12.260879 73.62228
## 107 -29.51267834 63.11222 12.951268 76.06349
## 108 -33.73323188 51.63443 13.373323 65.00776
## 109 -56.65111234 73.14060 15.665111 88.80571
## 110 -8.34353915 58.48017 10.834354 69.31453
## 111 -41.20912498 59.98742 14.120912 74.10833
## 112 -59.93841814 74.68376 15.993842 90.67760
## 113 -11.66554000 48.75910 11.166554 59.92565
## 114 -35.89986802 46.22514 13.589987 59.81513
## 115 -12.69662333 48.40374 11.269662 59.67340
## 116 -29.18637647 58.21832 12.918638 71.13696
## 117 7.12403785 41.54641 9.287596 50.83400
## 118 -4.86343302 50.22172 10.486343 60.70807
## 119 3.22935841 41.71874 9.677064 51.39580
## 120 -59.59425142 64.42625 15.959425 80.38568
## 121 -37.68228019 57.05983 13.768228 70.82806
## 122 -12.58731102 55.31664 11.258731 66.57537
## 123 6.82869440 34.90126 9.317131 44.21839
## 124 0.77481151 45.36441 9.922519 55.28693
## 125 -31.36255421 54.24383 13.136255 67.38009
## 126 -58.97132924 58.66144 15.897133 74.55858
## 127 -19.65649931 44.85050 11.965650 56.81615
## 128 -32.15744648 56.81300 13.215745 70.02874
## 129 4.22558628 37.05369 9.577441 46.63114
## 130 -32.23023517 63.85078 13.223024 77.07381
## 131 -36.28070609 67.33413 13.628071 80.96220
## 132 -27.20014819 42.86177 12.720015 55.58179
## 133 -8.76055503 35.23736 10.876056 46.11342
## 134 -44.76933238 65.72411 14.476933 80.20104
## 135 6.52360004 44.56029 9.347640 53.90793
## 136 -41.71730737 69.34164 14.171731 83.51337
## 137 -57.78944673 57.88776 15.778945 73.66670
## 138 -15.89964513 43.49657 11.589965 55.08653
## 139 -52.77201054 71.40787 15.277201 86.68507
## 140 -27.59750016 43.38842 12.759750 56.14817
## 141 -39.19176306 63.80045 13.919176 77.71963
## 142 -55.43107678 64.27713 15.543108 79.82024
## 143 -12.64301677 54.60984 11.264302 65.87414
## 144 -31.92712251 55.12866 13.192712 68.32138
## 145 -33.01891655 54.81734 13.301892 68.11923
## 146 9.28548601 40.23451 9.071451 49.30596
## 147 -59.55827995 68.49818 15.955828 84.45400
## 148 -31.67747538 62.15352 13.167748 75.32127
## 149 -13.24989367 56.83064 11.324989 68.15563
## 150 -47.56645544 64.30023 14.756646 79.05688
## 151 -47.21242151 70.41130 14.721242 85.13254
## 152 -6.76043184 54.50287 10.676043 65.17891
## 153 -43.06804784 62.88993 14.306805 77.19674
## 154 5.14670834 36.73272 9.485329 46.21805
## 155 4.31938543 46.75000 9.568061 56.31806
## 156 -54.21455657 61.97504 15.421456 77.39650
## 157 -14.60880590 52.26467 11.460881 63.72555
## 158 -41.71951521 69.70082 14.171952 83.87277
## 159 -21.79087868 50.36988 12.179088 62.54897
## 160 -32.90969851 56.00217 13.290970 69.29314
## 161 -17.20024317 47.73043 11.720024 59.45046
## 162 -34.17880879 56.68408 13.417881 70.10196
## 163 -5.46857205 50.83254 10.546857 61.37940
## 164 -59.30827988 69.24897 15.930828 85.17980
## 165 -32.92551780 63.92415 13.292552 77.21670
## 166 6.37841911 53.31790 9.362158 62.68006
## 167 -37.71370603 59.40793 13.771371 73.17930
## 168 -45.48552862 58.75989 14.548553 73.30844
## 169 -25.08077784 48.48923 12.508078 60.99731
## 170 -18.57599769 59.41094 11.857600 71.26854
## 171 -43.60126090 56.89836 14.360126 71.25849
## 172 -15.32177647 53.64727 11.532178 65.17944
## 173 -8.98685715 49.72170 10.898686 60.62039
## 174 -5.90454477 51.26849 10.590454 61.85895
## 175 -11.82091762 48.98129 11.182092 60.16338
## 176 -43.87712404 58.99988 14.387712 73.38759
## 177 -24.37298220 55.67226 12.437298 68.10956
## 178 -42.66836758 51.87319 14.266837 66.14003
## 179 -58.25629032 73.44492 15.825629 89.27055
## 180 -1.18927109 39.58136 10.118927 49.70029
## 181 -45.21598673 68.31483 14.521599 82.83642
## 182 -7.03706222 48.83176 10.703706 59.53547
## 183 -6.53129105 45.16240 10.653129 55.81553
## 184 -37.73464354 62.94566 13.773464 76.71912
## 185 -16.80767700 60.27289 11.680768 71.95366
## 186 5.64553464 46.88441 9.435447 56.31985
## 187 -45.32101245 61.95208 14.532101 76.48418
## 188 -5.92341356 45.71238 10.592341 56.30473
## 189 -32.17893511 54.30301 13.217894 67.52090
## 190 -48.73614143 68.57484 14.873614 83.44845
## 191 1.94683065 41.53411 9.805317 51.33942
## 192 -15.31951525 38.73594 11.531952 50.26790
## 193 -7.58430392 48.32015 10.758430 59.07858
## 194 9.11124597 25.06228 9.088875 34.15116
## 195 -1.64302519 54.04527 10.164303 64.20957
## 196 -27.13041464 49.76046 12.713041 62.47350
## 197 -51.90527450 55.85741 15.190527 71.04794
## 198 -22.66812008 63.04635 12.266812 75.31316
## 199 -33.02267992 60.49694 13.302268 73.79921
## 200 -1.81201306 42.26369 10.181201 52.44489
## 201 -59.10314685 60.13295 15.910315 76.04326
## 202 -23.60609813 59.67025 12.360610 72.03085
## 203 -36.95962268 53.72159 13.695962 67.41755
## 204 -33.88772937 67.60982 13.388773 80.99859
## 205 -34.41350969 50.36535 13.441351 63.80670
## 206 -44.04290082 64.60189 14.404290 79.00618
## 207 -8.09209591 56.16776 10.809210 66.97697
## 208 -6.35932414 46.23121 10.635932 56.86714
## 209 -44.44137452 57.10823 14.444137 71.55237
## 210 -3.89402793 47.14033 10.389403 57.52973
## 211 -3.64800721 39.39551 10.364801 49.76031
## 212 -17.74043798 47.98719 11.774044 59.76124
## 213 -44.27400504 62.57074 14.427401 76.99814
## 214 -7.65562134 58.50079 10.765562 69.26636
## 215 -30.17592035 50.61069 13.017592 63.62828
## 216 -1.43590003 46.10327 10.143590 56.24686
## 217 -5.49223091 47.74640 10.549223 58.29562
## 218 8.01820702 30.62032 9.198179 39.81850
## 219 5.92705271 49.52418 9.407295 58.93148
## 220 -56.94888618 66.12955 15.694889 81.82444
## 221 -41.02503433 56.73607 14.102503 70.83858
## 222 -40.37130262 57.13442 14.037130 71.17155
## 223 5.05344884 35.94580 9.494655 45.44046
## 224 -14.78871153 52.88525 11.478871 64.36412
## 225 -47.98961684 65.56372 14.798962 80.36268
## 226 9.29865772 50.66008 9.070134 59.73021
## 227 -43.69982315 69.61211 14.369982 83.98209
## 228 2.25918813 43.42085 9.774081 53.19493
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## 581 35.89942685 30.72487 44.359771 75.08464
## 582 43.98057467 16.50790 47.592230 64.10013
## 583 32.06807322 32.36654 42.827229 75.19377
## 584 33.65743455 28.34876 43.462974 71.81173
## 585 7.87211966 47.22702 33.148848 80.37587
## 586 53.55097944 18.71051 51.420392 70.13091
## 587 -4.32633458 38.11425 28.269466 66.38372
## 588 8.44929126 45.51312 33.379717 78.89284
## 589 43.52809025 28.83488 47.411236 76.24611
## 590 59.11220655 19.88144 53.644883 73.52632
## 591 1.90160365 45.28138 30.760641 76.04202
## 592 50.04747638 22.68894 50.018991 72.70793
## 593 9.87856765 30.41989 33.951427 64.37132
## 594 43.91809738 27.68080 47.567239 75.24804
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## 596 9.86287826 46.70705 33.945151 80.65220
## 597 -8.80052667 50.46932 26.479789 76.94911
## 598 3.41502681 33.80569 31.366011 65.17170
## 599 57.31210146 35.04772 52.924841 87.97256
## 600 12.04855803 34.03158 34.819423 68.85101
## 601 52.89146739 12.11107 51.156587 63.26766
## 602 20.70649138 37.80636 38.282597 76.08896
## 603 -7.55878811 52.67140 26.976485 79.64788
## 604 3.98502730 48.66239 31.594011 80.25640
## 605 32.80855564 30.47999 43.123422 73.60341
## 606 8.23796102 41.30790 33.295184 74.60309
## 607 15.86202360 37.12426 36.344809 73.46907
## 608 54.16979013 15.38328 51.667916 67.05120
## 609 29.03590285 24.12614 41.614361 65.74050
## 610 21.87049779 33.07549 38.748199 71.82369
## 611 58.95299231 18.89330 53.581197 72.47450
## 612 -9.36371965 45.61928 26.254512 71.87380
## 613 -3.79726234 43.43213 28.481095 71.91322
## 614 11.92417952 47.75631 34.769672 82.52598
## 615 33.22342111 22.07879 43.289368 65.36816
## 616 43.32460817 32.81237 47.329843 80.14221
## 617 43.66936617 23.42984 47.467746 70.89759
## 618 16.98113379 27.61750 36.792454 64.40995
## 619 48.43302465 29.77777 49.373210 79.15098
## 620 5.67530554 46.04081 32.270122 78.31093
## 621 14.73703343 35.31608 35.894813 71.21089
## 622 27.37697503 27.43686 40.950790 68.38765
## 623 59.26839940 18.75486 53.707360 72.46222
## 624 34.62107363 34.78995 43.848429 78.63838
## 625 21.76325471 22.97721 38.705302 61.68252
## 626 5.21959852 48.35522 32.087839 80.44306
## 627 33.19470026 33.37621 43.277880 76.65409
## 628 1.14733244 40.39758 30.458933 70.85651
## 629 57.56880772 17.32634 53.027523 70.35386
## 630 17.26261782 38.25942 36.905047 75.16446
## 631 -5.58568343 47.52531 27.765727 75.29104
## 632 58.16354734 10.56708 53.265419 63.83250
## 633 -6.38844788 48.37911 27.444621 75.82373
## 634 20.95010608 31.25045 38.380042 69.63050
## 635 47.89656926 31.48513 49.158628 80.64375
## 636 7.68608901 47.43613 33.074436 80.51056
## 637 25.03187798 32.89341 40.012751 72.90617
## 638 2.11967479 34.98403 30.847870 65.83190
## 639 16.57894959 28.26014 36.631580 64.89172
## 640 57.97412096 16.64649 53.189648 69.83614
## 641 42.30355075 29.68518 46.921420 76.60660
## 642 3.90254059 36.40497 31.561016 67.96599
## 643 33.63438446 25.35702 43.453754 68.81077
## 644 47.53616796 30.35751 49.014467 79.37198
## 645 3.16716294 40.72653 31.266865 71.99340
## 646 23.22769728 30.98708 39.291079 70.27816
## 647 -1.93167000 56.58382 29.227332 85.81115
## 648 24.14656985 42.44701 39.658628 82.10563
## 649 5.46969261 45.47182 32.187877 77.65970
## 650 47.73631030 25.48685 49.094524 74.58137
## 651 8.25067643 36.43451 33.300271 69.73478
## 652 44.59000673 19.90466 47.836003 67.74066
## 653 31.99263197 24.48198 42.797053 67.27903
## 654 44.07889126 18.48648 47.631557 66.11804
## 655 45.39232774 19.68365 48.156931 67.84058
## 656 55.45707325 27.83656 52.182829 80.01939
## 657 18.14014748 32.64077 37.256059 69.89683
## 658 3.37135000 41.06917 31.348540 72.41771
## 659 22.05532273 36.21930 38.822129 75.04143
## 660 42.74127647 19.95659 47.096511 67.05310
## 661 48.87835826 13.26050 49.551343 62.81184
## 662 31.77763424 32.00582 42.711054 74.71688
## 663 16.73386235 47.75627 36.693545 84.44982
## 664 24.59739088 40.55435 39.838956 80.39331
## 665 39.83212549 29.40185 45.932850 75.33470
## 666 58.67813909 21.13169 53.471256 74.60294
## 667 51.58862381 19.44275 50.635450 70.07820
## 668 55.49573886 21.32192 52.198296 73.52021
## 669 36.04467641 25.90380 44.417871 70.32167
## 670 20.82412381 43.57516 38.329650 81.90481
## 671 47.36963803 22.74003 48.947855 71.68789
## 672 5.27841176 47.93622 32.111365 80.04759
## 673 58.87774927 19.14430 53.551100 72.69540
## 674 34.42798660 29.58571 43.771195 73.35690
## 675 -9.26136531 56.92906 26.295454 83.22451
## 676 14.03755100 45.29930 35.615020 80.91432
## 677 14.66853627 34.89059 35.867415 70.75801
## 678 47.80103873 29.86743 49.120415 78.98784
## 679 -9.25467122 52.44936 26.298132 78.74749
## 680 11.67253510 32.96692 34.669014 67.63593
## 681 -7.74790903 52.38677 26.900836 79.28760
## 682 48.30087868 28.45022 49.320351 77.77057
## 683 21.88425941 27.21915 38.753704 65.97286
## 684 25.00511796 35.88493 40.002047 75.88698
## 685 7.37011492 43.38629 32.948046 76.33434
## 686 -5.68984228 47.31351 27.724063 75.03757
## 687 15.35310206 37.93269 36.141241 74.07394
## 688 31.74762579 26.68322 42.699050 69.38227
## 689 20.52438653 31.49863 38.209755 69.70838
## 690 0.64950381 44.01811 30.259802 74.27792
## 691 35.67126193 22.15612 44.268505 66.42463
## 692 20.75400034 35.80015 38.301600 74.10175
## 693 5.62298873 41.62736 32.249195 73.87656
## 694 -3.83620255 46.30170 28.465519 74.76722
## 695 29.09357766 24.39511 41.637431 66.03254
## 696 25.99445041 36.04329 40.397780 76.44107
## 697 38.42120922 23.99033 45.368484 69.35881
## 698 4.58612879 42.48695 31.834452 74.32140
## 699 26.06878121 35.45288 40.427512 75.88039
## 700 7.97834821 52.22699 33.191339 85.41833
## 701 50.96789549 28.40828 50.387158 78.79544
## 702 13.33324696 42.51851 35.333299 77.85181
## 703 21.90062279 32.10952 38.760249 70.86977
## 704 56.57396160 19.75320 52.629585 72.38278
## 705 11.21869300 41.17839 34.487477 75.66587
## 706 53.14576863 22.38858 51.258307 73.64689
## 707 2.72762839 43.38095 31.091051 74.47200
## 708 38.14783913 31.86851 45.259136 77.12764
## 709 23.04247289 25.37411 39.216989 64.59110
## 710 13.73824611 34.90440 35.495298 70.39970
## 711 13.17459572 36.79820 35.269838 72.06804
## 712 21.80312146 47.70712 38.721249 86.42837
## 713 54.48839070 22.19812 51.795356 73.99348
## 714 45.81171695 26.34944 48.324687 74.67412
## 715 21.24144557 48.41302 38.496578 86.90960
## 716 57.68317798 28.61101 53.073271 81.68428
## 717 42.68098362 17.17243 47.072393 64.24482
## 718 31.08456767 36.34688 42.433827 78.78071
## 719 45.31076307 31.20175 48.124305 79.32605
## 720 56.28718998 26.11055 52.514876 78.62542
## 721 53.81911605 25.52738 51.527646 77.05503
## 722 7.39054061 45.52632 32.956216 78.48254
## 723 46.84575806 29.14874 48.738303 77.88704
## 724 11.20084615 40.37352 34.480338 74.85386
## 725 22.42337598 34.85869 38.969350 73.82805
## 726 11.92155826 38.31859 34.768623 73.08721
## 727 20.56562671 33.52759 38.226251 71.75384
## 728 0.76861706 36.83630 30.307447 67.14375
## 729 45.31251092 26.59662 48.125004 74.72163
## 730 50.59893285 15.99256 50.239573 66.23214
## 731 0.45133606 42.83337 30.180534 73.01391
## 732 40.65552778 30.59642 46.262211 76.85863
## 733 51.51828443 26.49494 50.607314 77.10225
## 734 47.18381443 19.74098 48.873526 68.61451
## 735 10.60760694 38.33194 34.243043 72.57499
## 736 1.50788975 37.81458 30.603156 68.41774
## 737 55.91953926 16.01915 52.367816 68.38697
## 738 4.18923121 43.40281 31.675692 75.07850
## 739 43.79396285 25.72404 47.517585 73.24163
## 740 47.88156301 27.42515 49.152625 76.57777
## 741 10.42343448 37.64021 34.169374 71.80959
## 742 -4.11416220 45.99015 28.354335 74.34449
## 743 39.26618472 22.94351 45.706474 68.64998
## 744 56.47545044 16.04139 52.590180 68.63157
## 745 21.19317922 36.16399 38.477272 74.64126
## 746 56.39356842 25.36893 52.557427 77.92636
## 747 57.62312955 17.72395 53.049252 70.77320
## 748 36.76316727 29.24046 44.705267 73.94573
## 749 55.09533692 26.91598 52.038135 78.95411
## 750 17.83891072 43.51162 37.135564 80.64719
## 751 -9.72895248 51.08255 26.108419 77.19097
## 752 29.85677850 24.19996 41.942711 66.14267
## 753 -5.25861227 54.20675 27.896555 82.10330
## 754 -2.62733766 44.81473 28.949065 73.76379
## 755 22.95083637 26.35341 39.180335 65.53374
## 756 5.39161910 45.18902 32.156648 77.34567
## 757 26.01163372 31.91711 40.404653 72.32177
## 758 -1.53087209 45.15772 29.387651 74.54537
## 759 18.12139976 41.96061 37.248560 79.20917
## 760 42.96867194 20.12164 47.187469 67.30911
## 761 41.91950636 29.94806 46.767803 76.71586
## 762 22.19133318 45.62168 38.876533 84.49822
## 763 30.74303202 40.61340 42.297213 82.91061
## 764 44.32063950 23.66449 47.728256 71.39275
## 765 55.06142558 13.84438 52.024570 65.86895
## 766 33.70968845 30.43673 43.483875 73.92060
## 767 -5.40894299 44.33706 27.836423 72.17348
## 768 5.23825202 36.01729 32.095301 68.11259
## 769 31.79323794 39.62629 42.717295 82.34359
## 770 57.20311417 25.31550 52.881246 78.19674
## 771 3.66398494 39.92792 31.465594 71.39351
## 772 26.99224444 33.75158 40.796898 74.54848
## 773 31.29451483 27.92125 42.517806 70.43906
## 774 24.22564007 39.50323 39.690256 79.19349
## 775 37.81657222 35.42892 45.126629 80.55555
## 776 20.21312258 31.45439 38.085249 69.53964
## 777 13.69635518 39.05774 35.478542 74.53628
## 778 13.23125153 46.29698 35.292501 81.58948
## 779 -2.96820177 36.06896 28.812719 64.88168
## 780 38.83378592 26.78337 45.533514 72.31688
## 781 20.16483037 37.75621 38.065932 75.82214
## 782 52.50342017 21.95976 51.001368 72.96112
## 783 26.28344134 35.07488 40.513377 75.58825
## 784 56.62151238 16.71615 52.648605 69.36476
## 785 54.34863060 16.52957 51.739452 68.26903
## 786 30.61046475 34.69808 42.244186 76.94226
## 787 -1.22207134 30.88638 29.511171 60.39755
## 788 38.37788459 28.11931 45.351154 73.47047
## 789 9.14753620 43.08459 33.659014 76.74360
## 790 46.47984608 22.27451 48.591938 70.86645
## 791 36.40346398 30.24272 44.561386 74.80410
## 792 16.20886554 35.75921 36.483546 72.24276
## 793 5.98902110 45.43634 32.395608 77.83195
## 794 19.96463847 31.25168 37.985855 69.23753
## 795 35.88686411 30.11201 44.354746 74.46676
## 796 -8.29841420 44.62998 26.680634 71.31062
## 797 53.01167665 15.16361 51.204671 66.36828
## 798 57.23840809 19.68635 52.895363 72.58171
## 799 8.82574371 34.21870 33.530297 67.74899
## 800 4.56045402 40.83599 31.824182 72.66017
## 801 33.04756163 33.90167 43.219025 77.12069
## 802 20.23729327 30.72583 38.094917 68.82075
## 803 33.09828518 32.79795 43.239314 76.03726
## 804 -9.34423092 51.92597 26.262308 78.18828
## 805 43.69684457 19.37831 47.478738 66.85705
## 806 -8.62874476 42.21737 26.548502 68.76587
## 807 50.18825757 34.23967 50.075303 84.31497
## 808 -2.74923346 46.78099 28.900307 75.68130
## 809 -6.78410432 32.08069 27.286358 59.36705
## 810 -4.90593438 45.39735 28.037626 73.43498
## 811 44.59317972 22.45571 47.837272 70.29298
## 812 7.13524492 46.39274 32.854098 79.24684
## 813 29.36500014 35.09294 41.746000 76.83894
## 814 38.87157642 32.16453 45.548631 77.71316
## 815 10.42126806 44.89201 34.168507 79.06052
## 816 44.28437265 21.50605 47.713749 69.21980
## 817 6.73023629 39.92067 32.692095 72.61277
## 818 51.21069626 31.08641 50.484279 81.57069
## 819 7.27374071 31.97576 32.909496 64.88525
## 820 59.53907798 27.57964 53.815631 81.39527
## 821 26.08745474 37.16427 40.434982 77.59925
## 822 0.18644586 53.50090 30.074578 83.57548
## 823 -8.52893173 50.67350 26.588427 77.26193
## 824 35.63516752 23.49787 44.254067 67.75194
## 825 27.71136963 39.56448 41.084548 80.64903
## 826 45.98743412 23.89485 48.394974 72.28983
## 827 13.87985671 40.13132 35.551943 75.68326
## 828 23.22829692 43.22203 39.291319 82.51335
## 829 -5.93092650 38.04577 27.627629 65.67340
## 830 -1.43104848 51.37173 29.427581 80.79931
## 831 7.55588254 46.80942 33.022353 79.83178
## 832 38.93511956 20.86836 45.574048 66.44241
## 833 39.29548071 29.02638 45.718192 74.74457
## 834 26.39776989 30.26871 40.559108 70.82782
## 835 26.97179480 30.25831 40.788718 71.04702
## 836 36.77901249 32.77944 44.711605 77.49104
## 837 2.84836391 50.86728 31.139346 82.00662
## 838 46.97002536 23.23553 48.788010 72.02354
## 839 4.20985384 40.78051 31.683942 72.46446
## 840 26.92385239 28.91135 40.769541 69.68089
## 841 -0.15169964 37.81900 29.939320 67.75832
## 842 57.85779033 16.23407 53.143116 69.37719
## 843 34.64888760 29.48244 43.859555 73.34200
## 844 11.66816287 46.40075 34.667265 81.06802
## 845 34.68646865 32.30113 43.874587 76.17572
## 846 21.47188594 40.07615 38.588754 78.66491
## 847 -4.46130480 47.58384 28.215478 75.79932
## 848 59.00068807 17.06703 53.600275 70.66730
## 849 33.70789161 27.86916 43.483157 71.35232
## 850 20.60185028 41.47064 38.240740 79.71138
## 851 44.64247840 38.48638 47.856991 86.34337
## 852 35.25224620 23.21236 44.100898 67.31325
## 853 3.90358876 40.61215 31.561436 72.17359
## 854 9.38879679 34.90259 33.755519 68.65810
## 855 26.60399366 34.38923 40.641597 75.03083
## 856 41.33236441 34.80813 46.532946 81.34108
## 857 48.94764075 19.79005 49.579056 69.36910
## 858 54.88408126 26.59124 51.953633 78.54487
## 859 17.05293376 34.88652 36.821174 71.70769
## 860 41.78318934 36.59477 46.713276 83.30804
## 861 22.68645743 41.17437 39.074583 80.24895
## 862 16.35150558 30.61402 36.540602 67.15462
## 863 35.93922174 27.08265 44.375689 71.45834
## 864 30.19617431 31.52115 42.078470 73.59962
## 865 52.76925183 23.50494 51.107701 74.61264
## 866 36.31799410 36.41909 44.527198 80.94629
## 867 57.19360809 15.67198 52.877443 68.54943
## 868 16.19069478 43.43378 36.476278 79.91006
## 869 25.59940638 32.49900 40.239763 72.73877
## 870 4.51388458 40.98800 31.805554 72.79355
## 871 -5.56265387 40.41075 27.774938 68.18569
## 872 37.98331320 31.79675 45.193325 76.99007
## 873 48.27705164 26.74678 49.310821 76.05760
## 874 41.72914782 30.28198 46.691659 76.97364
## 875 4.12670328 48.46257 31.650681 80.11325
## 876 38.27295686 33.25832 45.309183 78.56751
## 877 0.92381279 41.61665 30.369525 71.98617
## 878 53.51516958 16.62168 51.406068 68.02775
## 879 33.03164609 23.05200 43.212658 66.26466
## 880 0.03016346 47.19642 30.012065 77.20848
## 881 36.80788981 24.03996 44.723156 68.76311
## 882 -5.26876849 48.50404 27.892493 76.39653
## 883 49.29346108 33.47121 49.717384 83.18860
## 884 38.69345384 28.51835 45.477382 73.99573
## 885 53.35903740 20.22864 51.343615 71.57226
## 886 23.95671318 25.97501 39.582685 65.55770
## 887 58.99367850 24.12381 53.597471 77.72128
## 888 26.64732747 30.51966 40.658931 71.17859
## 889 38.98181339 34.42363 45.592725 80.01636
## 890 24.71826857 30.88222 39.887307 70.76952
## 891 50.70018713 31.48477 50.280075 81.76485
## 892 36.36638942 34.94235 44.546556 79.48890
## 893 32.63239437 34.22036 43.052958 77.27331
## 894 2.63083118 36.74777 31.052332 67.80010
## 895 18.43415412 27.02362 37.373662 64.39728
## 896 31.82795837 36.02330 42.731183 78.75448
## 897 -2.74241145 42.41454 28.903035 71.31757
## 898 59.73181562 15.25998 53.892726 69.15271
## 899 40.94520740 23.14727 46.378083 69.52535
## 900 37.72422433 31.17298 45.089690 76.26267
## 901 42.21568067 32.01866 46.886272 78.90494
## 902 26.83041406 26.77632 40.732166 67.50848
## 903 35.43787366 32.12870 44.175149 76.30385
## 904 1.90628891 47.22102 30.762516 77.98354
## 905 38.31260780 29.94314 45.325043 75.26818
## 906 31.01956729 34.26794 42.407827 76.67577
## 907 -9.80684887 43.20964 26.077260 69.28690
## 908 39.32568977 28.68301 45.730276 74.41329
## 909 52.24686890 20.51676 50.898748 71.41551
## 910 40.90772962 21.83249 46.363092 68.19558
## 911 28.98280300 33.99475 41.593121 75.58787
## 912 54.17878126 17.16892 51.671513 68.84043
## 913 32.10790131 29.37164 42.843161 72.21480
## 914 54.63678406 17.34716 51.854714 69.20188
## 915 -9.04105458 50.84280 26.383578 77.22638
## 916 2.95583759 40.56978 31.182335 71.75211
## 917 5.65147389 39.48817 32.260590 71.74876
## 918 42.92435663 27.16771 47.169743 74.33745
## 919 53.72941694 22.72208 51.491767 74.21384
## 920 51.86737113 19.64785 50.746948 70.39480
## 921 25.82485060 36.14906 40.329940 76.47900
## 922 25.54569174 43.81366 40.218277 84.03193
## 923 56.94678385 20.26746 52.778714 73.04617
## 924 52.75352698 28.82017 51.101411 79.92158
## 925 47.53786727 32.12464 49.015147 81.13979
## 926 3.13531956 49.44198 31.254128 80.69611
## 927 49.07622840 18.10466 49.630491 67.73515
## 928 19.38015102 41.09908 37.752060 78.85114
## 929 19.80056661 26.86976 37.920227 64.78999
## 930 32.84491949 29.80049 43.137968 72.93846
## 931 -3.48142360 47.59833 28.607431 76.20576
## 932 -2.65123198 49.80337 28.939507 78.74288
## 933 40.30445281 26.32183 46.121781 72.44361
## 934 22.35197746 31.82111 38.940791 70.76190
## 935 -0.61394783 45.08844 29.754421 74.84286
## 936 1.51007709 39.23008 30.604031 69.83411
## 937 14.64076282 33.11418 35.856305 68.97048
## 938 21.98762474 33.45349 38.795050 72.24854
## 939 4.82871603 45.26007 31.931486 77.19155
## 940 3.88749967 34.71150 31.555000 66.26650
## 941 -2.93955205 39.87854 28.824179 68.70272
## 942 51.19039149 18.59577 50.476157 69.07193
## 943 51.51981481 24.39418 50.607926 75.00211
## 944 39.12508001 23.09569 45.650032 68.74573
## 945 32.47448948 27.51672 42.989796 70.50651
## 946 53.68943040 27.28845 51.475772 78.76422
## 947 26.51746927 29.01549 40.606988 69.62248
## 948 46.98920819 17.70710 48.795683 66.50278
## 949 4.30882137 43.01295 31.723529 74.73648
## 950 26.46387181 33.11213 40.585549 73.69768
## 951 15.33481458 39.53522 36.133926 75.66914
## 952 -0.54129589 49.65739 29.783482 79.44088
## 953 -2.93809571 32.94181 28.824762 61.76657
## 954 10.96300656 44.30219 34.385203 78.68739
## 955 1.24477323 52.25154 30.497909 82.74945
## 956 -5.82493194 36.89550 27.670027 64.56553
## 957 56.50617960 16.33668 52.602472 68.93915
## 958 42.87602603 23.23409 47.150410 70.38450
## 959 1.08046291 41.02908 30.432185 71.46127
## 960 32.52673153 25.40959 43.010693 68.42028
## 961 32.51417364 32.67341 43.005669 75.67908
## 962 49.39539607 20.45756 49.758158 70.21572
## 963 22.52472840 30.53633 39.009891 69.54623
## 964 33.65286434 39.88796 43.461146 83.34910
## 965 27.99203684 40.02379 41.196815 81.22061
## 966 7.02586779 44.20511 32.810347 77.01546
## 967 8.38256863 32.25039 33.353027 65.60341
## 968 52.66292045 32.23835 51.065168 83.30351
## 969 57.42030602 26.96205 52.968122 79.93018
## 970 5.07706083 41.24060 32.030824 73.27143
## 971 17.78635252 40.71748 37.114541 77.83202
## 972 52.19044432 21.38435 50.876178 72.26053
## 973 39.90379125 21.31785 45.961517 67.27937
## 974 11.45691508 37.50110 34.582766 72.08387
## 975 40.39186597 27.92679 46.156746 74.08353
## 976 40.38593643 29.94846 46.154375 76.10284
## 977 6.84485605 43.84536 32.737942 76.58330
## 978 7.31594340 47.97663 32.926377 80.90301
## 979 4.22766691 41.80365 31.691067 73.49472
## 980 7.01145383 39.68193 32.804582 72.48651
## 981 23.64225085 29.79352 39.456900 69.25042
## 982 50.91978614 21.14641 50.367914 71.51432
## 983 7.89268345 34.58878 33.157073 67.74586
## 984 24.91744559 34.09453 39.966978 74.06151
## 985 33.10992153 23.25717 43.243969 66.50113
## 986 47.36384627 30.82294 48.945539 79.76848
## 987 49.73978462 15.74087 49.895914 65.63678
## 988 15.93774317 33.72547 36.375097 70.10057
## 989 33.24353847 34.94126 43.297415 78.23868
## 990 -9.27076773 50.07140 26.291693 76.36309
## 991 24.90495807 39.88410 39.961983 79.84608
## 992 12.13074249 43.94438 34.852297 78.79668
## 993 44.29683099 19.49528 47.718732 67.21401
## 994 14.45683000 34.19584 35.782732 69.97857
## 995 -6.29156080 52.99687 27.483376 80.48025
## 996 12.37184240 32.09865 34.948737 67.04738
## 997 55.33845170 26.15127 52.135381 78.28665
## 998 -3.35818426 47.79305 28.656726 76.44977
## 999 41.40925018 27.92381 46.563700 74.48751
## 1000 39.81406235 30.66882 45.925625 76.59445
## above_cutoff below_cutoff
## 1 FALSE TRUE
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## 997 TRUE FALSE
## 998 FALSE TRUE
## 999 TRUE FALSE
## 1000 TRUE FALSE
We could manually run the first stage model, generate predicted tutoring and then use those predicted values in the second stage model like we did in the instrumental variables example, but that’s tedious and nobody wants to do all that work. We’ll use iv_robust() from the estimatr package instead.
Note: be careful! if there are missing values (in D), you may fail to replicate the fuzzy effect by hand sum(is.na(crab_data$fished)) #ok for the simulated data
Instead, we need to run a 2SLS model that includes our instrument in the first stage, which will then remove the endogeneity built into participation in the program. We’ll estimate this set of models:
##NEED TO FINISHED - IS THIS RIGHT??? \[ \begin{aligned} \widehat{\text{Fished}} &= \gamma_0 + \gamma_1 \text{Size}_\text{centered} + \gamma_2 \text{Below cutoff} + \omega \\ \text{Fishing Mortality} &= \beta_0 + \beta_1 \text{Size}_\text{centered} + \beta_2 \widehat{\text{Fished}} + \epsilon \end{aligned} \]
We can also use nonparametric methods to measure the size of the fuzzy gap at the cutoff. We’ll use rdrobust(). T
To do fuzzy estimation with rdrobust(), use the fuzzy argument to specify the treatment column (or fished in our case). Importantly (and confusingly! this took me waaaaay too long to figure out!), you do not need to specify an instrument (or even create one!). All you need to specify is the column that indicates treatment status—rdrobust() will do all the above/below-the-cutoff instrument stuff behind the scenes for you.
##NEED TO FINISHED WITH IV CREATED AS ENTRANCE EXAM
rdrobust(y = crab_data$fishing_mortality, x = crab_data$size_centered,
c = 0, fuzzy = crab_data$fished) %>%
summary()
## Fuzzy RD estimates using local polynomial regression.
##
## Number of Obs. 1000
## BW type mserd
## Kernel Triangular
## VCE method NN
##
## Number of Obs. 480 520
## Eff. Number of Obs. 121 142
## Order est. (p) 1 1
## Order bias (q) 2 2
## BW est. (h) 8.850 8.850
## BW bias (b) 17.518 17.518
## rho (h/b) 0.505 0.505
## Unique Obs. 480 520
##
## First-stage estimates.
##
## =============================================================================
## Method Coef. Std. Err. z P>|z| [ 95% C.I. ]
## =============================================================================
## Conventional 0.117 0.156 0.750 0.453 [-0.188 , 0.422]
## Robust - - 1.143 0.253 [-0.143 , 0.543]
## =============================================================================
##
## Treatment effect estimates.
##
## =============================================================================
## Method Coef. Std. Err. z P>|z| [ 95% C.I. ]
## =============================================================================
## Conventional 20.944 1.687 12.413 0.000 [17.637 , 24.251]
## Robust - - 10.491 0.000 [16.939 , 24.722]
## =============================================================================
That’s all! Using nonparametric methods, with a triangular kernel and a bandwidth of ±12.96, the causal effect of the tutoring program for compliers in the bandwidth is 9.683.
We can (and should!) do all the other nonparametric robustness checks that we talked about in the regression discontinuity example, like modifying the bandwidth (ideal, half, double) and messing with the kernel (uniform, triangular, Epanechnikov) to see how robust the finding is. But again, we won’t do any of that here.
https://dimewiki.worldbank.org/Regression_Discontinuity
the MixTape by Scott Cummingham: https://mixtape.scunning.com/06-regression_discontinuity
RDD with interactions: https://www.jepusto.com/rdd-interactions/
Modified from the MixTape by Scott Cummingham: https://mixtape.scunning.com/06-regression_discontinuity
library(tidyverse)
# simulate the data
dat <- tibble(
x = rnorm(1000, 50, 25)
) %>%
mutate(
x = if_else(x < 0, 0, x)
) %>%
filter(x < 100)
# cutoff at x = 70, no jump
dat <- dat %>%
mutate(
D = if_else(x > 70, 1, 0),
y1 = 25 + 0 * D + 1.5 * x + rnorm(n(), 0, 20)
)
ggplot(aes(x, y1, colour = factor(D)), data = dat) +
geom_point(alpha = 0.5) +
geom_vline(xintercept = 70, colour = "grey", linetype = 2)+
stat_smooth(method = "lm", se = F) +
labs(x = "Test score (X)", y = "Potential Outcome (Y1)")
## `geom_smooth()` using formula = 'y ~ x'
## simulate the discontinuity
dat <- dat %>%
mutate(
y2 = 25 + 40 * D + 1.5 * x + rnorm(n(), 0, 20)
)
# figure
ggplot(aes(x, y2, colour = factor(D)), data = dat) +
geom_point(alpha = 0.5) +
geom_vline(xintercept = 70, colour = "grey", linetype = 2) +
stat_smooth(method = "lm", se = F) +
labs(x = "Test score (X)", y = "Potential Outcome (Y)")
## `geom_smooth()` using formula = 'y ~ x'
# simultate nonlinearity
dat <- tibble(
x = rnorm(1000, 100, 50)
) %>%
mutate(
x = case_when(x < 0 ~ 0, TRUE ~ x),
D = case_when(x > 140 ~ 1, TRUE ~ 0),
x2 = x*x,
x3 = x*x*x,
y3 = 10000 + 0 * D - 100 * x + x2 + rnorm(1000, 0, 1000)
) %>%
filter(x < 280)
ggplot(aes(x, y3, colour = factor(D)), data = dat) +
geom_point(alpha = 0.2) +
geom_vline(xintercept = 140, colour = "grey", linetype = 2) +
stat_smooth(method = "lm", se = F) +
labs(x = "Test score (X)", y = "Potential Outcome (Y)")
## `geom_smooth()` using formula = 'y ~ x'
ggplot(aes(x, y3, colour = factor(D)), data = dat) +
geom_point(alpha = 0.2) +
geom_vline(xintercept = 140, colour = "grey", linetype = 2) +
stat_smooth(method = "loess", se = F) +
labs(x = "Test score (X)", y = "Potential Outcome (Y)")
## `geom_smooth()` using formula = 'y ~ x'
–>
We can check the size two different ways: parametrically (i.e. using lm() with specific parameters and coefficients), and nonparametrically (i.e. not using lm() or any kind of straight line and instead drawing lines that fit the data more precisely). We’ll do it both ways.
First we’ll do it parametrically by using linear regression. Here we want to explain the variation in final scores based on the entrance exam score and participation in the tutoring program:
\[ \text{Exit exam} = \beta_0 + \beta_1 \text{Entrance exam score}_\text{centered} + \beta_2 \text{Tutoring program} + \epsilon \]
Instead of using linear regression to measure the size of the discontinuity, we can use nonparametric methods. Essentially this means that R will not try to fit a straight line to the data—instead it’ll curve around the points and try to fit everything as smoothly as possible.
The rdrobust() function makes it really easy to measure the gap at the cutoff with nonparametric estimation. Here’s the simplest version: